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\begin_layout Title
Language Learning Diary - Part Two
\end_layout
\begin_layout Date
February-August 2021
\end_layout
\begin_layout Author
Linas Vepstas
\end_layout
\begin_layout Abstract
The language-learning effort involves research and software development
to implement the ideas concerning unsupervised learning of grammar, syntax
and semantics from corpora.
This document contains supplementary notes and a loosely-organized semi-chronol
ogical diary of results.
The notes here might not always makes sense; they are a short-hand for
my own benefit, rather than aimed at you, dear reader!
\end_layout
\begin_layout Section*
Introduction
\end_layout
\begin_layout Standard
Part two of the diary on the language-learning effort has three major sections.
These are summarized here.
This part has been closed, and part three has been started (there will
be no further updates here.)
\end_layout
\begin_layout Subsubsection*
Experiments with random grammars.
Feb-May 2021.
\end_layout
\begin_layout Standard
Attempt to run the full learning calibration pipeline on a randomly-generated
grammar.
This pipeline creates a random grammar, creates a random corpus with this
grammar, runs the learning pipeline, learns a grammar, and then compares
the learned grammar to the original one.
\end_layout
\begin_layout Standard
The goal of doing this is to be able to perform
\begin_inset Quotes eld
\end_inset
calibration
\begin_inset Quotes erd
\end_inset
: by comparing the input and output grammars for fidelity, we could explore
the parameter space, and find the best tuning for the learning pipeline.
This sounds good in principle, it failed in practice.
\end_layout
\begin_layout Standard
Doing this entailed numerous bug fixes and some new code to get everything
working (so that's a good thing).
A bit of interesting mathematics popped up: merging is a non-commutative
operation.
When merging words into a word class, one must also merge connectors into
connector classes.
The initial idea was that merges can be done as (quasi-)linear operations:
Attached to each word is a vector, and creating a word-class is as simple
as summing vectors.
Or possibly intersecting the basis elements, and then summing only those
parts where the basis overlap (thus, the quasi-linear merge.
This is needed, because non-overlapping vectors correspond to different
word senses, and we need for factor those out.
\end_layout
\begin_layout Standard
One might imagine that merging connectors is as simple as creating vectors
for those, the so-called
\begin_inset Quotes eld
\end_inset
shape
\begin_inset Quotes erd
\end_inset
vectors, and summing those.
Because of how shapes are created, one might think that one could merge
words, then shapes, or first shapes and only then words, and get the same
thing.
Not so: the merge operation is non-commutative.
The result depends on the order it's done in.
This is interesting, but troublesome.
\end_layout
\begin_layout Standard
The goal of merging is to find two things that are similar, and then place
them in a common class.
Since the
\begin_inset Quotes eld
\end_inset
things
\begin_inset Quotes erd
\end_inset
are word-disjunct pairs, one wishes to find some unambiguous, unique way
of merging them.
This seems not to be the case: there is some kind of
\begin_inset Quotes eld
\end_inset
torsion
\begin_inset Quotes erd
\end_inset
or
\begin_inset Quotes eld
\end_inset
curvature
\begin_inset Quotes erd
\end_inset
in the system.
It's not clear how to even characterize this concept clearly.
An unambiguous merge seems to be asking for some kind of flattening or
abelianization of the merge procedure.
It's not clear how to do this.
\end_layout
\begin_layout Standard
Mathematically, the formalities are still opaque.
The notation for word-disjunct pairs is awkward.
The notation for the linearizations are awkward.
When I think about linearization, it feels very geometric: as if one is
working with some tangent space to some manifold.
But there's no manifold, and the analogies quickly falter.
Progress requires a much clearer, a much more concrete description of what's
going on with this network.
\end_layout
\begin_layout Standard
The experiments ended with confusion.
Besides the above non-commutative aspect to merging, it became clear that
many different initial grammars are effectively equivalent, in that they
generate similar, if not identical corpora.
Thus, the naive idea of calibration fails: one can't just learn the input
grammar; at best one can learn the class of that grammar.
But, as of now, we don't have any clear way of defining a
\begin_inset Quotes eld
\end_inset
class of equivalent grammars
\begin_inset Quotes erd
\end_inset
, or more broadly,
\begin_inset Quotes eld
\end_inset
a class of very similar grammars
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Standard
A secondary problem was that the random grammar didn't resemble a natural
language grammar.
That was not the intent.
It was hoped that by picking a Zipfian distribution of vocabulary words,
a Zipf distribution of disjuncts, throwing in some synonyms into the mix,
the result would resemble a natural language.
This was not the case.
Despite having done a lot of distributional studies of English, it seems
that I still don't know how to characterize a natural language very well.
I don't know how to measure the complexity of a grammar, or the complexity
of a corpus.
There are formal questions: how can one tell when two grammars are isomorphic?
Almost isomorphic? There are practical questions: what is the best way
to factor the word-disjunct matrix?
\end_layout
\begin_layout Standard
In the end, it all got muddy, with now easy, obvious way of moving forward.
\end_layout
\begin_layout Subsubsection*
Back to English.
July 2021
\end_layout
\begin_layout Standard
The advantage of working with English is that one can develop a gut-feel
as to whether one is making progress.
It's hard to look at a random grammar, and decide
\begin_inset Quotes eld
\end_inset
does that make sense?
\begin_inset Quotes erd
\end_inset
So, back to English.
Earlier English experiments went so swimmingly well, that surely moving
forward should be easy!
\end_layout
\begin_layout Standard
An embarrassing problem was immediately encountered: Setting the merge threshold
at some apparently innocuous value, that would result in similar words
getting merged immediately resulted in chaos and failure.
This is very unlike the earlier experiments, where things went quite very
well.
Something went wrong.
What is it?
\end_layout
\begin_layout Standard
Well, several things.
\end_layout
\begin_layout Itemize
Merging requires a merge threshold.
Earlier experiments searched for this threshold carefully, and then applied
it.
Here, we complacently picked one, based on past experience and gut-sense,
and it was set too low.
\end_layout
\begin_layout Itemize
Earlier merges used an overlap strategy, of assuming that similar words
might have different disjunct sets just because some words were just too
thinly sampled, to have seen such disjuncts.
This seems like a good idea, but is perhaps naive.
If one word-disjunct pair has been observed 100 times, and given another,
similar word, that word-disjunct pair has been observed zero times, well,
that might not be due to under-sampling.
100 is really different than zero, and assigning a fraction of the 100
to the merge cluster is perhaps a bad idea.
There is a semi-formal proof that merging in this way will
\emph on
always
\emph default
reduce the MI.
So if the goal is entropy maximization, the overlap merge strategy is a
bad idea.
\end_layout
\begin_layout Itemize
Some confusion regarding the above ensues.
The MI between words is explored, and compared to the MI between merged
classes.
After a lot of work, it really does become clear that overlap merging always
makes grammatical class MI's worse, and that a pure intersection merge
always maximizes MI.
\end_layout
\begin_layout Itemize
It was expected that using Shapes would improve similarity scores.
After some work, it seems that this is not the case - they mostly seem
to make things worse.
It's not at all clear which this is so.
Shapes should have helped, but they don't!
\end_layout
\begin_layout Itemize
To keep calculations relatively fast, trimmed datasets were used.
It now seems that trimming damages the data.
This is still a problem.
Trimming seems like it should help learning, but again, it seems like it
doesn't.
The causes of this are still not intuitively clear.
\end_layout
\begin_layout Standard
The biggest thing that went wrong was that, when using MI as a measure of
similarity, some rather dis-similar words had a high MI.
How can that be? Turns out that if one compares a very large vector (a
word with lots of disjuncts, i.e.
a word with a large support) to another word with a very short vector (a
word with only one disjunct, for example), the result can be a very high
MI.
Yet this is obviously not what we want: short vectors should not be similar
to long vectors.
This prompts an exploration of other kinds of similarity measures.
\end_layout
\begin_layout Subsubsection*
Similarity Smackdown, July-August 2021
\end_layout
\begin_layout Standard
The poor merge results prompted a step back, to judge different similarity
measures (metrics).
Twelve
\emph on
ad hoc
\emph default
\begin_inset Quotes eld
\end_inset
part of speech
\begin_inset Quotes erd
\end_inset
clusters were defined.
They were
\emph on
ad hoc
\emph default
in the sense of being hand-picked out of the first few thousand words with
the highest observation counts.
The various similarity measures were used to computer the inter-cluster
and intra-cluster similarity.
The quality of the metrics was assessed based on their ability to discriminate
between clusters, by taking the ratio of the RMS sizes of the clusters
to the RMS cluster separations.
Higher discrimination is judged as being better.
\end_layout
\begin_layout Standard
Nine different similarity measures were evaluated:
\end_layout
\begin_layout Itemize
Mutual information
\end_layout
\begin_layout Itemize
Joint probability
\end_layout
\begin_layout Itemize
Variation of information
\end_layout
\begin_layout Itemize
Normalized variation of information
\end_layout
\begin_layout Itemize
Cosine distance
\end_layout
\begin_layout Itemize
Log cosine distance
\end_layout
\begin_layout Itemize
Vector overlap (plain Jaccard)
\end_layout
\begin_layout Itemize
Conditional Jaccard distance
\end_layout
\begin_layout Itemize
Probability Jaccard distance.
\end_layout
\begin_layout Standard
These were compared one datasets with and without shapes, at seven different
levels of trimming (so, 14 datasets total).
The results were as follows:
\end_layout
\begin_layout Itemize
Trimming is consistently deleterious to results.
However, using the full dataset blows up both RAM and compute time.
The best compromise, for the given datasets, appear to be to discard all
words that have a small number of observation counts, and to discard sections
with an observation count of less than two.
\end_layout
\begin_layout Itemize
Shapes mostly fail to improve similarity measurements, for any of the similarity
measures.
Using shapes makes all vectors more similar (smaller RMS intra-cluster,
but also smaller RMS inter-cluster).
The overall effect of shapes is mixed: they seem to improve things only
for highly-trimmed datasets.
Given that shapes add a large computational burden, they seem to offer
no benefit.
(They sharply increase the amount of time needed to compute similarity,
and can triple the RAM usage.)
\end_layout
\begin_layout Itemize
The best overall similarity measure was vector overlap.
Second place overall goes to Conditional Jaccard (which uses probabilities
in the Jaccard min/max formula) Third place goes to Probability Jaccard.
MI is consistently in fifth place.
\end_layout
\begin_layout Standard
A few comments.
Although the motivation for these experiments was the large MI between
short and long vectors, the experiments actually eliminated short vectors.
That is, the hand-picked clusters consisted of words with relatively high
counts, and thus relatively long vectors.
\end_layout
\begin_layout Standard
The failure of Probability Jaccard to dominate is also surprising.
The Wikipedia article explains that it is in some sense optimal: it provides
\begin_inset Quotes eld
\end_inset
maximally consistent sampling
\begin_inset Quotes erd
\end_inset
.
This appears not to help us.
It is also very computationally intensive; its the slowest similarity measure
of all.
\end_layout
\begin_layout Section*
February 2021 - Closed Loop Calibration
\end_layout
\begin_layout Standard
Part two of the diary on the language-learning effort starts with the new
task of of closed loop learning.
The idea of closed loop learning is that the accuracy of the learned grammars
can be very closely monitored and measured, thus allowing the learning
algorithms to be tuned for speed and measured for accuracy.
The closed loop is a basic five-step process:
\end_layout
\begin_layout Enumerate
Generating random but controlled grammars
\end_layout
\begin_layout Enumerate
Generate a text corpus from these grammars
\end_layout
\begin_layout Enumerate
Learn a new grammar from the corpus
\end_layout
\begin_layout Enumerate
Compare the controlled-grammar to the learned-grammar
\end_layout
\begin_layout Enumerate
Tune algorithms and procedures, and repeat.
\end_layout
\begin_layout Standard
The step-by-step instructions can be found in the file
\begin_inset Quotes eld
\end_inset
README-Calibration.md
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Section*
February 2021
\end_layout
\begin_layout Standard
Restart the project, finally making some headway.
ToDo items:
\end_layout
\begin_layout Itemize
Describe how the
\begin_inset Quotes eld
\end_inset
uniform random sampling of sentences
\begin_inset Quotes erd
\end_inset
is performed.
\end_layout
\begin_layout Itemize
Enable weighted random sampling of sentences.
\end_layout
\begin_layout Itemize
Fix multi-sense bug in gen-dict.scm circa line 85
\end_layout
\begin_layout Standard
First run.
Instructions in
\begin_inset Quotes eld
\end_inset
README-Calibration.md
\begin_inset Quotes erd
\end_inset
.
Lots and lots of bugs fixed and lots of pipeline was automated.
\end_layout
\begin_layout Section*
March 2021
\end_layout
\begin_layout Standard
First real experiment, in expt-7/expt-8/expt-9.
The dict in expt-7 fails to generate the correct corpus, because the generator
does not expand synonyms (this is a combinatorial explosion and it just
doesn't do that.) Expanded by hand in expt-8.
Same data, new scripts and config in expt-9.
\end_layout
\begin_layout Standard
Issues:
\end_layout
\begin_layout Itemize
After MST parsing, the grammar is correct, in that (I think) it will produce
exactly the same corpus.
However, the rules are different (and more verbose).
TODO: check that the same corpus is actually produced.
How? Answer: generate the corpus, and compare...
\end_layout
\begin_layout Itemize
After MST (MPG) parsing, the verbs link to obj-determiner instead of object.
Why? Was there a cutoff that was missed? The corpus is just ..
tiny.
\end_layout
\begin_layout Itemize
After (gram-classify-greedy-discrim 0.5 4) the right clusters are produced,
but the connectors are not clustered; they need to be.
TBD.
\end_layout
\begin_layout Itemize
Result: the correct corpus is produced (via manual checking), however, extra
sentences are produced, which are missing the verb.
This is because the wall links to the two determiners, and the wall can
be skipped.
However, the sentence with the verb has a higher MI.
(11.51 instead of 8.92.
See below.) This is due to a bug.
See below.
\end_layout
\begin_layout Standard
Here's the result:
\begin_inset VSpace defskip
\end_inset
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
linkparser> the squirrel a dog
\end_layout
\begin_layout Plain Layout
Found 1 linkage (1 had no P.P.
violations)
\end_layout
\begin_layout Plain Layout
Unique linkage, cost vector = (UNUSED=0 DIS=-8.92 LEN=2)
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
+-----------TB-----------+
\end_layout
\begin_layout Plain Layout
+---TB---+---TE---+ +-TE-+
\end_layout
\begin_layout Plain Layout
| | | | |
\end_layout
\begin_layout Plain Layout
LEFT-WALL.2 the.1 squirrel.3 a.1 dog.3
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
linkparser> the squirrel saw a dog
\end_layout
\begin_layout Plain Layout
Found 1 linkage (1 had no P.P.
violations)
\end_layout
\begin_layout Plain Layout
Unique linkage, cost vector = (UNUSED=0 DIS=-11.51 LEN=4)
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
+--------------TB--------------+
\end_layout
\begin_layout Plain Layout
| +-----TC-----+
\end_layout
\begin_layout Plain Layout
+---TB---+---TE---+---TF--+-TD-+-TE-+
\end_layout
\begin_layout Plain Layout
| | | | | |
\end_layout
\begin_layout Plain Layout
LEFT-WALL.2 the.1 squirrel.3 saw.4 a.1 dog.3
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
Hypothesis: The MI of wall-verb is lower than the MI of wall-determiner.
Thus, the planar parser always picks the wall-determiner.
Lets find out.
\end_layout
\begin_layout Standard
Heh.
There is no MI wall-verb! Ouch.
This was due to bad sampling; fixed github.com/opencog/opencog commit 895226228
Mar 16 2021.
Sheesh.
\end_layout
\begin_layout Subsection*
Expt-10
\end_layout
\begin_layout Standard
expt-10, this is fixed.
The parse trees are now very rich.
Generated sentences:
\end_layout
\begin_layout Itemize
length 4 - none
\end_layout
\begin_layout Itemize
length 5 - the expected ones.
\end_layout
\begin_layout Itemize
6 - LEFT-WALL LEFT-WALL plus valid sentence
\end_layout
\begin_layout Itemize
7 - none
\end_layout
\begin_layout Itemize
8 - LEFT-WALL the mouse saw the dog chased a bird
\end_layout
\begin_layout Itemize
9 - double left wall
\end_layout
\begin_layout Itemize
10 - the LEFT-WALL a cat saw the dog chased a squirrel – and also a triple-left-
wall.
\end_layout
\begin_layout Standard
WTF.
what's with the crazy multi-left-wall!? Heh.
Here we go:
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
+---------------------TI-------------------+
\end_layout
\begin_layout Plain Layout
+------------------TB-----------------+ |
\end_layout
\begin_layout Plain Layout
| +------TI-----+ | |
\end_layout
\begin_layout Plain Layout
| +---TB--+--TC-+--TO--+-TE-+-TC-+
\end_layout
\begin_layout Plain Layout
| | | | | | |
\end_layout
\begin_layout Plain Layout
LEFT-WALL.2 LEFT-WALL.2 a.1 mouse.5 saw.3 a.1 cat.5
\end_layout
\end_inset
\end_layout
\begin_layout Subsection*
Expt-11
\end_layout
\begin_layout Standard
So...
expt-11 places a period at the end of every sentence.
That terminates the infinite-recursive lengths being generated to only
finite-length sentences.
There is a total of three different parses.
All have exactly the same cost.
These are as follows:
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
Found 3 linkages (3 had no P.P.
violations) Linkage
1, cost vector = (UNUSED=0 DIS=-15.67 LEN=10)
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
+-------------------TH-------------------+
\end_layout
\begin_layout Plain Layout
| +------------TJ-----------+
\end_layout
\begin_layout Plain Layout
+------TG------+ +----TE----+
\end_layout
\begin_layout Plain Layout
+---TF---+--TD-+---TI--+--TC--+--TD-+-TJ-+
\end_layout
\begin_layout Plain Layout
| | | | | | |
\end_layout
\begin_layout Plain Layout
LEFT-WALL.2 the.1 dog.3 chased.4 the.1 cat.3 ..5
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
Linkage 2, cost vector = (UNUSED=0 DIS=-15.67 LEN=11)
\end_layout
\begin_layout Plain Layout
+-------------------TH-------------------+
\end_layout
\begin_layout Plain Layout
| +------------TJ-----------+
\end_layout
\begin_layout Plain Layout
+------TG------+------TB------+----TE----+
\end_layout
\begin_layout Plain Layout
+---TF---+--TD-+---TI--+--TC--+--TD-+-TJ-+
\end_layout
\begin_layout Plain Layout
| | | | | | |
\end_layout
\begin_layout Plain Layout
LEFT-WALL.2 the.1 dog.3 chased.4 the.1 cat.3 ..5
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
Linkage 3, cost vector = (UNUSED=0 DIS=-15.67 LEN=12)
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
+-------------------TH-------------------+
\end_layout
\begin_layout Plain Layout
| +------------TJ-----------+
\end_layout
\begin_layout Plain Layout
| | +--------TK-------+
\end_layout
\begin_layout Plain Layout
+------TG------+ | +----TE----+
\end_layout
\begin_layout Plain Layout
+---TF---+--TD-+---TI--+--TC--+--TD-+-TJ-+
\end_layout
\begin_layout Plain Layout
| | | | | | |
\end_layout
\begin_layout Plain Layout
LEFT-WALL.2 the.1 dog.3 chased.4 the.1 cat.3 ..5
\end_layout
\end_inset
\begin_inset VSpace defskip
\end_inset
Notable in the above:
\end_layout
\begin_layout Itemize
Only one parse, the third one, links the main verb to a wall.
And then its the right wall, not the left wall.
\end_layout
\begin_layout Itemize
The determiners seem to be over-linked, and judged to play a too-important
role.
\end_layout
\begin_layout Itemize
The output grammar is much more highly detailed and constrained than the
intended grammar.
\end_layout
\begin_layout Standard
Questions:
\end_layout
\begin_layout Itemize
What happens if there are a lot more verbs? Would this make the determiners
less important, more important, or have no effect? (My guess is
\begin_inset Quotes eld
\end_inset
no effect
\begin_inset Quotes erd
\end_inset
)
\end_layout
\begin_layout Itemize
To downgrade the importance of determiners would seem to require having
sentences without them in it.
\end_layout
\begin_layout Standard
TBD:
\end_layout
\begin_layout Itemize
Waiting on completion of link-generator so that multiple-sense corpora can
be generated.
(enabled in lg pull req #1175) Or something like that ...
what is the right strategy here? Need to rethink to avoid combinatorial
explosion, while also verifying category contents.
\end_layout
\begin_layout Itemize
Fix bug to allow multiple-sense word definitions in multiple dict locations.
\end_layout
\begin_layout Itemize
Dict generation should auto-handle placing a period at the end of the sentence.
\end_layout
\begin_layout Subsection*
Expt-12
\end_layout
\begin_layout Standard
Start work on a single-sense random dictionary.
Hit assorted issues with the scripts.
\end_layout
\begin_layout Standard
Results:
\end_layout
\begin_layout Itemize
non-classified dict has 134391 disjuncts, 11 uni-classes including left-wall.
These are raw disjuncts.
\end_layout
\begin_layout Itemize
classified dict (i.e.
that on which grammatical classification has been performed) has 11286
disjuncts
\end_layout
\begin_layout Standard
Time to generate 50 sentences, and all possible sentences, in seconds.
\end_layout
\begin_layout Standard
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Data is semi-meaningless, scripts were broken, data processing would start
before data was fully loaded.
Try again.
Upon restart, the number of sentences generated is order of magnitude lower.
Presumably due to corrected clustering; above clustered incomplete lists
of disjuncts and thus over-generalized.
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Expt-13, expt-14, expt-15
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Try again with the same initial corpus.
Expt-13 overflowed with fake warning message, so I couldn't see the log;
thus expt-14 is an exact rerun.
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Exact
\begin_inset Quotes erd
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in the sense of using the same config.
However, the random sampling of pairs during pair counting was different.
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Then expt-15 uses exactly the same corpus, with a period at the end of sentence
placed manually.
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\end_layout
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num: number of sentences generated (expt-13)
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expt-14: number of sentences generated
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\begin_layout Itemize
corpus: number of sentences in input corpus.
Capped at 25K sentences for the longer sentences.
Second number is how many could have been generated.
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expt-15: redo, but with a period at the end of the sentence.
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Expt-13 and expt-14 differ only in how the pair-counts were collected (they
are randomly different samplings of pairs).
Both wildly over-sample the corpus.
\end_layout
\begin_layout Standard
The expt-13/14 input corpus lacks periods at the ends of sentences.
This seems to be the most likely explanation for the over-generation; i.e.
last time a period was lacking, the same thing happened.
So expt-15 takes exactly the same corpus - identical copy, and adds a period.
This does sharply cut down on the number of sentences, especially the long
ones, but still over-generates.
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General processing stats:
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Issues:
\end_layout
\begin_layout Itemize
There seems to be a dataset issue: both the disjunct pairs and gram pairs
have 1/3rd of them without counts on them.
..
They are not being saved, after clustering (clustering causes deletion
of many disjuncts, and alteration of counts on all disjuncts.) I'm guessing
this failure results in bad MI's? Anyway, its a bug is fixed in the new
clustering shell scripts.
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After above fix, verify export.
gram-1 is a re-export of the original expt-13 run, while gram-4 is export
of the fixed run.
(Both start with the same disjuncts.
I think there's nothing stochastic/random during processing, so it should
be repeatable...)
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11.57=11.29+1.09-MI
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So ..
similar but not the same.
How about sentence generation? expt-13-gram-1 and expt-13-gram-1a are identical
(the 1a version is from a re-export; so we're exporting the same stuff).
\end_layout
\begin_layout Standard
The classes in gram-1 are <b f> <e j> and left-wall.
The classes in gram-4 are the same plus <i#uni> ...
there was no word i in gram-1 !! Wow, that's a big drop.
\end_layout
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\align center
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time for all
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\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
OK, so the numbers are dramatically larger.
Apparently, this is due to the previously dropped word i.
Yikes!
\end_layout
\begin_layout Standard
Well, the above is massively under-counting – it is only sampling one random
word-draw per class.
Since multiple words are in each class...
\end_layout
\begin_layout Standard
...
anyway, this is nuts, because the connectors need to be classified, instead
of issuing new connector types.
So more work before something meaningful is possible.
\end_layout
\begin_layout Subsection*
expt-13 vs expt-15 precision, recall
\end_layout
\begin_layout Standard
So now that we've got things working, lets look at precision and recall.
Clearly precision will be terrible, but maybe recall will be excellent?
Compare expt-13-gram-4 to expt-15-gram-2 (which rebuilds after fixing the
borked save.
\end_layout
\begin_layout Standard
Uhh ..
No its not working yet, in the sense that the connector classes are being
mishandled.
Those need to be grouped correctly before export.
More work...
\end_layout
\begin_layout Subsection*
expt-16
\end_layout
\begin_layout Standard
Due to absence of left-wall in the above, breaking the dict-compare step,
tried again, generating a new dict with walls (after manually adding a
wall to the dict of expt-15, and ending punctuation to the dictionary.)
Well ...
clustering worked quite differently.
Here's a summary.
\end_layout
\begin_layout Standard
Pair counting seems to be more-or-less the same, slightly higher MI=0.030
which is still minuscule.
MPG entropy, counts etc.
look similar to the earlier runs.
\end_layout
\begin_layout Standard
Gram classification: only 2 words assigned to the same class.
Oh, this used the
\begin_inset Quotes eld
\end_inset
disinfo
\begin_inset Quotes erd
\end_inset
classifier, whereas the earlier runs used the
\begin_inset Quotes eld
\end_inset
fuzz
\begin_inset Quotes erd
\end_inset
classifier.
That could account for everything, I guess.
Lets take a look.
\end_layout
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\end_inset
\end_layout
\begin_layout Standard
\align center
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11 x 75552
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gram entropy
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11.48=10.01+3.15-MI
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10.38=9.94+1.90-MI
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10.45=9.91+2.06-MI
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\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
Clearly, the resulting clusters are sensitive to the parameters controlling
classification.
The above parameters seemed reasonable for the large English dataset.
They may be unreasonable here!? But this is very unclear.
\end_layout
\begin_layout Standard
The MI's are larger, across the board (vs.
0.8 or 0.9 before.) How about sentence generation? We expect disinfo to be
more accurate, since it did very little clustering.
\end_layout
\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
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\begin_layout Standard
Again, the discrim is under-counting, because of more categorization.
\end_layout
\begin_layout Standard
Next step: fix the conjoined clustering, with shapes.
\end_layout
\begin_layout Standard
Wow.
So MM^T entropy with shapes is 5.03 which is huge compared to the dj-only
MM^T so it really is something new and different! With shapes, and with
gram-disinfo, there were no merges.
\end_layout
\begin_layout Standard
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11 x 75552
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1011444
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11.48=10.01+3.15-MI
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\begin_layout Standard
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OK, so looks like shape created no categories at all.
So how does that work out for generation?
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
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link-generator -l learned -c 123123123 -s 3
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\end_layout
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time for all
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> 25K
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\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
OK, so wildly over-generating sentences, despite effectively no clustering
being done.
Didn't we do an experiment without clustering?? I can't find it above.
Why are we over-generating? How to best explain it? Too small a vocabulary?
\end_layout
\begin_layout Itemize
Issue 1: given a fake-lang the generator is failing to generate all possible
sentences.
Fixed in link-grammar pull req #1175
\end_layout
\begin_layout Itemize
Issue 2: there are
\begin_inset Quotes eld
\end_inset
accidental
\begin_inset Quotes erd
\end_inset
synonyms cause of 1 above: many POS'es are shared in common between many
words but are not completely sampled, thus creating
\begin_inset Quotes eld
\end_inset
accidental synonyms
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Standard
OK, so back to that one, except this time, its in the shape of a bug...
\end_layout
\begin_layout Subsection*
Sections and disjuncts
\end_layout
\begin_layout Standard
The Sections that were learned in expt-16 have a surprising number of connectors
on them, averaging at 7.7 connectors per section.
This seems way too large.
What's up with that? Step one: get a more detailed view.
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
# column 1 length
\end_layout
\begin_layout Plain Layout
# column 2: number of Section
\end_layout
\begin_layout Plain Layout
# column 3: number of ConnectorSeq
\end_layout
\begin_layout Plain Layout
#
\end_layout
\begin_layout Plain Layout
1 0 0
\end_layout
\begin_layout Plain Layout
2 606 115
\end_layout
\begin_layout Plain Layout
3 2311 780
\end_layout
\begin_layout Plain Layout
4 3723 1904
\end_layout
\begin_layout Plain Layout
5 5150 4066
\end_layout
\begin_layout Plain Layout
6 8387 8183
\end_layout
\begin_layout Plain Layout
7 13907 13896
\end_layout
\begin_layout Plain Layout
8 17362 17362
\end_layout
\begin_layout Plain Layout
9 15043 15043
\end_layout
\begin_layout Plain Layout
10 9061 9061
\end_layout
\begin_layout Plain Layout
11 4000 4000
\end_layout
\begin_layout Plain Layout
12 1257 1257
\end_layout
\begin_layout Plain Layout
13 0
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\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Standard
So ...
long disjuncts appear on one and only one word (as witnessed by identical
counts for length 8 and above).
Short disjuncts might be shared with multiple words.
This implies that words are not mergeable, or rather, the distinct long
sections are preventing merger.
Here a graph of above.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename images-diary-two/dj-len.eps
width 80col%
\end_inset
\end_layout
\begin_layout Standard
So its very Gaussian, both peak and tails.
All these gaussians here and in earlier results imply that central-limit
theorems hold.
And all kinds of other classical theorems should hold.
Have not been leveraging those theorems, so far.
If we did, what would we get? Note it's Gaussian despite the fact that
negative connector-seq lengths don't make sense! Right? What would a negative
connector-seq length be? A repulsion? an anti-connect statement?
\begin_inset Quotes eld
\end_inset
Must never connect
\begin_inset Quotes erd
\end_inset
?
\end_layout
\begin_layout Standard
What happens if we exclude Sections with a low observation count?
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename images-diary-two/dj-len-cut.eps
width 50col%
\end_inset
\end_layout
\begin_layout Standard
The above shows all sections, and sections that were observed 5 or more
times.
Apparently, the long, complicated disjuncts are observed very rarely.
\end_layout
\begin_layout Standard
How should this be interpreted? Common sense seems to suggest that low-count
observations are
\begin_inset Quotes eld
\end_inset
noise
\begin_inset Quotes erd
\end_inset
and should be cut before any merging is performed.
Doing so will certainly increase the similarity of vectors.
But are cuts really needed? Perhaps the similarity measures can already
deal with these? If so, then the only reason for cuts would be performance,
rather than accuracy.
\end_layout
\begin_layout Standard
How do these affect cosine-similarity vs MI-similarity? Clearly, for cosine-simi
larity, low counts means short vector components, and so these will not
contribute much to the dot product.
Likewise, the MI-similarity is built on a dot-product, so again, these
should not contribute much.
Even more-so, since the MI-similarity never takes a square-root of the
dot product.
It does, however, re-weight basis elements in a fashion that I do not yet
have a good intuition for.
(Or rather, developed an intuition, and now I've forgotten what it was...
Hmmm.)
\end_layout
\begin_layout Standard
TODO: The Shapiro–Wilk test can be used to determine how close a distribution
is to a Gaussian.
The Wikipedia article on it is as clear as mud.
\end_layout
\begin_layout Section*
April-May 2021
\end_layout
\begin_layout Standard
Before starting expt-17 with fixed sampling, it is time to ponder connector
merging.
This is turning out to be non-trivial.
The general upshot of the below is that the original 2019 concept for Shapes
and CrossSections is a good, strong idea.
However, assorted non-commutativities arise, and how to resolve them appropriat
ely is not entirely clear.
Thus, we embark on a journey of discovery...
\end_layout
\begin_layout Standard
But first: connector merging is needed because without it, the clustering
fails to adequately reduce the size of dictionaries.
The meta question is this: if words A and B are determined to be near-synonyms,
and assigned to the same cluster, then what should be done with connectors
that have A or B in them? Should all such connectors be automatically replaced
by the cluster? The naive answer is,
\begin_inset Quotes eld
\end_inset
yes, they should be
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Standard
The less-naive response is
\begin_inset Quotes eld
\end_inset
but multiple word-senses
\begin_inset Quotes erd
\end_inset
.
That is, A and B are not just words, but are word-vectors, and vector B
may be a linear combination of two different word senses.
One of these senses might be synonymous to A, and the other might be completely
different.
Thus, we want to merge that part of vector B that is (nearly) co-linear
with A, while leaving behind a different vector B-prime that is associated
with some other word-sense for the word B.
When encountering B in a connector, is that B in the sense of wordclass-AB,
or is it in the sense of B-prime? If the former, then obviously, that connector
should be updated to use wordclass-AB; otherwise, it needs to be left alone,
thus implicitly defaulting to B-prime.
\end_layout
\begin_layout Standard
TBD: Writing the above, it appears that the need for an explicit B-prime
has been overlooked.
This could cause havoc, in allowing inappropriate linkages! This is important,
and needs to be addressed ...
\end_layout
\begin_layout Subsection*
Connector merging
\end_layout
\begin_layout Standard
Some of the questions that arise with connector merging, to figure out:
\end_layout
\begin_layout Itemize
How would connector merging affect clustering results?
\end_layout
\begin_layout Itemize
If connectors are merged, then how should vectors be handled? (Naive vectors
no longer work, because the basis is now different.)
\end_layout
\begin_layout Itemize
What is the best or most correct merge algorithm?
\end_layout
\begin_layout Standard
Lets look at a toy model.
Suppose the dict is
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
A: C+ & D+
\end_layout
\begin_layout Plain Layout
B: C+ & D+
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
C: A- & E+
\end_layout
\begin_layout Plain Layout
C: B- & F+
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
D: A- & G+
\end_layout
\begin_layout Plain Layout
D: B- & G+
\end_layout
\end_inset
\end_layout
\begin_layout Standard
From this, conclude that A and B can be merged.
However, the connectors on C cannot be merged, the connectors on D can
be merged.
The final dict is then (with some abuse of notation)
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
<wclass-AB>: C+ & D+
\end_layout
\begin_layout Plain Layout
A B: <wclass-AB>
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
C: A- & E+
\end_layout
\begin_layout Plain Layout
C: B- & F+
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
D: <wclass-AB>- & G+
\end_layout
\end_inset
Next suppose that we have
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
S: C+ & D+
\end_layout
\begin_layout Plain Layout
T: C+ & D+
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
C: S- & J+
\end_layout
\begin_layout Plain Layout
C: T- & F+
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
D: S- & H+
\end_layout
\begin_layout Plain Layout
D: T- & G-
\end_layout
\end_inset
The determination to merge is now trickier.
Naively, S can be merged into
\family typewriter
<wclass-AB>
\family default
but doing so would wreck the connector set on D, as it implies an
\family typewriter
(S- & G+)
\family default
which was not observed, and thus not mergeable.
...
unless we are willing to create such new unobserved cases.
\end_layout
\begin_layout Standard
Merging T looks OK.
\end_layout
\begin_layout Standard
Perhaps this looks bad because we are not doing shape vectors.
The shape vector for the above would be:
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
A: (C+ & D+) or (C:x- & E+) or (D:x- & G+)
\end_layout
\begin_layout Plain Layout
B: (C+ & D+) or (C:x- & F+) or (D:x- & G+)
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
C: (A- & E+) or ...
\end_layout
\begin_layout Plain Layout
C: (B- & F+) or ...
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
D: (A- & G+) or (A:C+ & x+)
\end_layout
\begin_layout Plain Layout
D: (B- & G+) or (B:C+ & x+)
\end_layout
\end_inset
\end_layout
\begin_layout Standard
So the decision to merge A and B is less clear-cut, when using shapes (they
are not perfect synonyms).
The shape variant of the S thing is
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
S: (C+ & D+) or (C:x- & J+) or (D:x- & H+)
\end_layout
\begin_layout Plain Layout
T: (C+ & D+) or (C:x- & F+) or (D:x- & G+)
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
D: (S- & H+) or (S:C+ & x+)
\end_layout
\begin_layout Plain Layout
D: (T- & H+) or (T:C+ & x+)
\end_layout
\end_inset
So using shapes weakens the decision to merge S into AB.
The decision to merge T remains strong.
So it seems that shapes do offer a stronger foundation on which to make
merge decisions.
They examine similarity (almost-synonymy) out to a greater distance from
the germ.
It seems like they also allow things to continue to be treated as vectors,
instead of muddling the concept of vectors.
After the AB-merge
\begin_inset Foot
status open
\begin_layout Plain Layout
When using the
\begin_inset Quotes eld
\end_inset
union-merge
\begin_inset Quotes erd
\end_inset
strategy, as described in in
\family typewriter
src/gram-projective.scm
\family default
.
In practice, the merge style used is typically
\family typewriter
merge-project
\family default
, which would accept only a fraction of
\family typewriter
(C:x- & E+)
\family default
and
\family typewriter
(C:x- & F+)
\family default
into the final vector.
\end_layout
\end_inset
the result would be:
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
<wclass-AB>: (C+ & D+) or (C:x- & E+) or (C:x- & F+) or (D:x- & G+)
\end_layout
\begin_layout Plain Layout
A B: <wclass-AB>
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
C: A- & E+
\end_layout
\begin_layout Plain Layout
C: B- & F+
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
D: (<wclass-AB>- & G+) or (<wclass-AB>:C+ & x+)
\end_layout
\end_inset
Now, given this merged-AB thing, when happens when we look at merging S?
Well, S is meh, T looks better.
The vector for T is comparable to the vector for <wclass-AB> so vector
similarity works for that merge decision.
\end_layout
\begin_layout Standard
Conclude: the original plan from a few years ago works and holds water.
Use shape vectors for merge decisions.
Once this is done, connectors can be swept up.
\end_layout
\begin_layout Subsection*
Non-commutativity
\end_layout
\begin_layout Standard
The above description pulls a sleight-of-hand, which presumes an algorithm
that is able to crawl across individual disjuncts, compare them, and update
words with merged word-classes.
Such an algorithm can be written (and has been written/prototyped).
It leads to some confusion, because the shapes/cross-sections are no longer
consistent.
Lets call the above the
\begin_inset Quotes eld
\end_inset
connector sweep algorithm
\begin_inset Quotes erd
\end_inset
, or
\begin_inset Quotes eld
\end_inset
sweep
\begin_inset Quotes erd
\end_inset
for short.
\end_layout
\begin_layout Standard
Starting with the above example:
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
<wclass-AB>: (C+ & D+) or (C:x- & E+) or (C:x- & F+) or (D:x- & G+)
\end_layout
\end_inset
the sections can be reconstructed from the cross-sections.
The reconstructed sections are
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
C: <wclass-AB>- & E+
\end_layout
\begin_layout Plain Layout
C: <wclass-AB>- & F+
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
D: <wclass-AB>- & G+
\end_layout
\end_inset
Comparing, the reconstructed section on D is the same as what the sweep
algo produced, but the sections on C are *not* what the sweep merge offers.
That is, the sweep is not commutative with the creation of shapes.
This is a problem for maintaining the consistency between sections and
cross-sections as clusters grow.
The lack of consistency will cause merge judgments to diverge...
\end_layout
\begin_layout Standard
Thus, we have at least two algorithms:
\end_layout
\begin_layout Itemize
Sweep-merge, as described above, where connectors are replaced by merged-connect
ors if and only if the the rest of the connector set is identical.
This merge algorithm is naively described, since it does not explain what
to do if there are multiple connectors in a connector set that might be
merged.
It's also naive in that it does not explain how counts (frequencies) are
to be handled.
\end_layout
\begin_layout Itemize
Reshape-merge, which performs the basic projective merge on the germ-vectors,
and then reconstructs Sections from CrossSections, thus restoring consistency
between sections and cross-sections.
It violates the intuitive correctness of the sweep-merge, but only perhaps
because the sweep-merge, as naively described above, assumed the
\begin_inset Quotes eld
\end_inset
union-merge
\begin_inset Quotes erd
\end_inset
strategy of transferring observation counts for vectors that are not perfectly
co-linear.
The projective-merge count transfers recognize the non-colinearity, and
obtain cluster centroids through weighting formulas.
\end_layout
\begin_layout Standard
The above ruminations suggest that reshape-merge enjoys an advantage over
sweep-merge, as it keeps the section/cross-section duality consistent.
\end_layout
\begin_layout Standard
Anyway, the original 2019 plan for using shapes seems to have been a good
plan.
\end_layout
\begin_layout Subsection*
Non-Commutativity, Again
\end_layout
\begin_layout Standard
The non-commutativity can be heightened with a slightly richer example.
Consider
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
A: (P+ & Q+) or (R+ & S+) or (K- & B+)
\end_layout
\begin_layout Plain Layout
B: (P+ & Q+) or (R+ & S+)
\end_layout
\begin_layout Plain Layout
C: B- & T+
\end_layout
\end_inset
The dictionary entry of C is present to remind us of the fact that, if B+
appears as a connector, then B- must also appear as a connector, somewhere.
\end_layout
\begin_layout Standard
Based on the first two sections, a decision might be made to merge, with
the count on
\family typewriter
(K- & B+)
\family default
being small enough that it does not disrupt the merge decision.
Expanding this into it's CrossSections, the full vectors are:
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
A: (P+ & Q+) or (R+ & S+) or (K- & B+)
\end_layout
\begin_layout Plain Layout
B: (P+ & Q+) or (R+ & S+) or (K- & A:x+) or (C:x- & T+)
\end_layout
\end_inset
The merge result is then
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
<wclass-AB>: (P+ & Q+) or (R+ & S+) or (K- & B+)
\end_layout
\begin_layout Plain Layout
<wclass-AB>: (K- & A:x+) or (C:x- & T+)
\end_layout
\end_inset
The cross-section leads to a reconstruction (reshape) of
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
A: K- & <wclass-AB>+
\end_layout
\begin_layout Plain Layout
C: <wclass-AB>- & T+
\end_layout
\end_inset
\end_layout
\begin_layout Standard
How is this to be interpreted? Let's explore some
\begin_inset Quotes eld
\end_inset
common sense
\begin_inset Quotes erd
\end_inset
reasoning.
\end_layout
\begin_layout Paragraph*
Case A:
\end_layout
\begin_layout Standard
The count on
\family typewriter
(K+ & B+)
\family default
is so small that it is considered to be noise, and is completely dropped
before merging even starts.
In this case, A and B are exactly co-linear (are exact synonyms).
The (naive) merge of A and B is completely unproblematic, except that it
leaves C without the ability to connect to anything.
This can be handled in one of two ways.
One way is to notice that, by detailed balance, the counts on this particular
C section must also be tiny, and so this section can be dropped from the
dictionary.
\end_layout
\begin_layout Standard
Another way to avoid this dangling-connector problem is to presume that
the dictionary also includes
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
D: L- & B+
\end_layout
\end_inset
which would provide a place for that bar B to connect.
but if this were the case, then we did the cross-sections on B wrong.
Fixing these would have given
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
<wclass-AB>: (P+ & Q+) or (R+ & S+) or (K- & B+)
\end_layout
\begin_layout Plain Layout
<wclass-AB>: (K- & A:x+) or (C:x- & T+) or (L- & D:x+)
\end_layout
\end_inset
which then reshapes to
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
A: K- & <wclass-AB>+
\end_layout
\begin_layout Plain Layout
C: <wclass-AB>- & T+
\end_layout
\begin_layout Plain Layout
D: L- & <wclass-AB>+
\end_layout
\end_inset
This is now fully linkable, and there are no dangling pure-B connectors.
\end_layout
\begin_layout Standard
In conclusion, this seems self-consistent either way: either we can drop
the
\family typewriter
A:(K+ & B+)
\family default
section entirely, and, by detailed balance, we can drop
\family typewriter
D:(L- & B+)
\family default
also; or we can keep both, and doing it correctly leaves nothing dangling.
\end_layout
\begin_layout Standard
Note that we have to be careful with tracking in the merge algo: when reshaping
to get the
\family typewriter
C:<wclass-AB>- & T+
\family default
section, we have to be careful to notice that the
\family typewriter
C:B- & T+
\family default
section was a donor, and so it should be removed (its counts driven to
zero).
Otherwise, C would end with both these sections on it, and it would be
a bit wonky.
\end_layout
\begin_layout Paragraph*
Case B:
\end_layout
\begin_layout Standard
The count on
\family typewriter
(K+ & B+)
\family default
is small but not ignorable.
It is small enough to not block the merge decision.
There are two issues to resolve.
The first is easy: the
\family typewriter
C:B- & T+
\family default
section should be recognized as a donor to
\family typewriter
C:<wclass-AB>- & T+
\family default
, and removed (its counts driven to zero).
This is easy enough to determine at the time of the merge.
\end_layout
\begin_layout Standard
The more difficult issue is what to do about the
\family typewriter
<wclass-AB>:(K- & B+)
\family default
section, which appears to have a dangling B connector, and the reshape
of
\family typewriter
A:(K- & <wclass-AB>+)
\family default
, which seems to be a double-count.
The first leaves a dangling A, the second leaves a dangling B.
It seems fairly clear that these should be harmonized, merged together,
to give
\family typewriter
<wclass-AB>:(K- & <wclass-AB>+)
\family default
.
It seems that this can be reasonably inferred and performed at the time
of creation, since the donors are readily identified.
\end_layout
\begin_layout Paragraph*
Case C:
\end_layout
\begin_layout Standard
The count on
\family typewriter
(K+ & B+)
\family default
is large, large enough to split.
That is, A should be understood to be the direct sum of two distinct word-sense
s, with one word-sense being
\family typewriter
<wclass-AB>
\family default
and the other being
\family typewriter
<A-prime>:(K+ & B+).
\family default
So, if were were able to be absolutely sure that A-prime was really a distinct
word-sense, then we should transfer none of the counts from the originating
section
\family typewriter
A:(K+ & B+)
\family default
to the
\family typewriter
<wclass-AB>
\family default
section.
\end_layout
\begin_layout Standard
So, starting with the vectors
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
A: (P+ & Q+) or (R+ & S+) or (K- & B+)[n]
\end_layout
\begin_layout Plain Layout
B: (P+ & Q+) or (R+ & S+) or (K- & A:x+)[n] or (C:x- & T+)
\end_layout
\end_inset
where square-bracket-n is the count on that section, we create a merge result
of the form
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
<wclass-AB>: (P+ & Q+) or (R+ & S+)
\end_layout
\begin_layout Plain Layout
<wclass-AB>: (C:x- & T+)
\end_layout
\begin_layout Plain Layout
<A-prime>: (K- & B+)[n]
\end_layout
\begin_layout Plain Layout
B: (K- & <A-prime>:x+)[n]
\end_layout
\end_inset
The cross-sections leads to a reconstruction (reshape) that appears to be
self-consistent, so I don't see any problems here.
\end_layout
\begin_layout Standard
If we were to split [n] into some fractional parts, then this would reduce
to a combination of case B and the current case, so that should also work.
\end_layout
\begin_layout Subsection*
Connector counts
\end_layout
\begin_layout Standard
(TBD, this section needs to be harmonized with the new text above...
the below was written before the above was rewritten...) Lets go through above
exercise, this time with counts.
Suppose the dict, with observation counts in square brackets, is
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
A: C+ & D+ [na]
\end_layout
\begin_layout Plain Layout
B: C+ & D+ [nb]
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
C: A- & E+ [nca]
\end_layout
\begin_layout Plain Layout
C: B- & F+ [ncb]
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
D: A- & G+ [nda]
\end_layout
\begin_layout Plain Layout
D: B- & G+ [ndb]
\end_layout
\end_inset
The shapes, with counts, are
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
A: (C+ & D+)[na] or (C:x- & E+)[nca] or (D:x- & G+)[nda]
\end_layout
\begin_layout Plain Layout
B: (C+ & D+)[nb] or (C:x- & F+)[ncb] or (D:x- & G+)[ndb]
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
C: (A- & E+)[nca] or ...
\end_layout
\begin_layout Plain Layout
C: (B- & F+)[ncb] or ...
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
D: (A- & G+)[nda] or (A:C+ & x+)[na]
\end_layout
\begin_layout Plain Layout
D: (B- & G+)[ndb] or (B:C+ & x+)[nb]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Merging A and B, with 100% of count transfer, gives
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
A B: (C+ & D+)[na+nb] or (C:x- & E+)[nca] or (C:x- & F+)[ncb] or (D:x-
& G+)[nda+ndb]
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
C: (A- & E+)[nca] or ...
\end_layout
\begin_layout Plain Layout
C: (B- & F+)[ncb] or ...
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
D: (A- & G+)[nda] or (A:C+ & x+)[na]
\end_layout
\begin_layout Plain Layout
D: (B- & G+)[ndb] or (B:C+ & x+)[nb]
\end_layout
\end_inset
Looking at the connectors on D, we see that they are mergeable, and that
the counts are consistent.
So, based on this toy model, we can either try to merge connectors directly,
or we can, at a later date, merge connectors by reconstructing them from
merged shapes.
Doing it either way should give the same counts: the operations are commutative.
\end_layout
\begin_layout Standard
This is not entirely obvious.
It seems to work for the toy example.
It seems like the toy example could be converted into a full proof.
Yet ...
are we missing something? Best bet is to write the code both ways, and
verify numerically that the operations are commutative.
\end_layout
\begin_layout Subsubsection*
Fractional counts
\end_layout
\begin_layout Standard
Lets try again, this time with fractional counts.
Suppose that instead of merging 100% of B into A, we merge only a fraction
\begin_inset Formula $0\le y\le1$
\end_inset
of the count.
This gives
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
A B: (C+ & D+)[na+y*nb] or (C:x- & E+)[nca] or (C:x- & F+)[y*ncb] or
(D:x- & G+)[nda+y*ndb]
\end_layout
\begin_layout Plain Layout
B: C+ & D+ [(1-y)nb] or (C:x- & F+)[(1-y)ncb] or (D:x- & G+)[(1-y)ndb]
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
C: (A- & E+)[nca] or ...
\end_layout
\begin_layout Plain Layout
C: (B- & F+)[ncb] or ...
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
D: (A- & G+)[nda] or (A:C+ & x+)[na]
\end_layout
\begin_layout Plain Layout
D: (B- & G+)[ndb] or (B:C+ & x+)[nb]
\end_layout
\end_inset
Then, apparently, the counts will be consistent if and only if the same
fraction is used when merging connectors.
\end_layout
\begin_layout Subsection*
Connector merging, with counts
\end_layout
\begin_layout Standard
To get all of the above correct, there is a series of unit tests.
They work well, but one of the more complex ones has become painfully difficult
to understand and debug.
It is reviewed here.
But first, a change of notation to make it more compact:
\end_layout
\begin_layout Itemize
The entry
\family typewriter
A:(B- & C+)
\family default
will be written as
\family typewriter
(A, BC)
\family default
.
Here, the parenthesis denote a pair (a matrix entry).
The letter sequence is just the connector sequence with the directional
indicators ignored.
\end_layout
\begin_layout Itemize
Entries with word-classes, such as
\family typewriter
<wclass-AB>:(K- & B+)
\family default
will be written as
\family typewriter
({AB},KB)
\family default
.
The word-class is denoted with set-notation curly braces.
Similarly,
\family typewriter
A:(K- & <wclass-AB>+)
\family default
will be written as
\family typewriter
(A,K{AB})
\family default
.
\end_layout
\begin_layout Itemize
Cross-sections, which were written above as
\family typewriter
D:(A:C+ & x+)
\family default
will be written as
\family typewriter
[D, <A, Cv>]
\family default
.
The angle brackets denote the shape, and the lower-case
\family typewriter
v
\family default
denotes the location of the variable in the connector sequence.
The square brackets just serve to remind that a cross-section is being
discussed.
\end_layout
\begin_layout Itemize
A property called
\begin_inset Quotes eld
\end_inset
detailed balance
\begin_inset Quotes erd
\end_inset
is introduced.
This is the idea that corresponding sections and cross-sections should
have exactly the same observation count on them.
Thus for example, given a section
\family typewriter
(A, BC)
\family default
which was observed N times, one expects that the two cross-sections derived
from it, namely
\family typewriter
[B, <A, vC>]
\family default
and
\family typewriter
[C, <A, Bv>]
\family default
are also both observed N times each.
Prior to any merging, detailed balance holds
\begin_inset Quotes eld
\end_inset
automatically
\begin_inset Quotes erd
\end_inset
or tautologically, as a trite statement about how counting is done.
The goal is that connector merging should preserve detailed balance as
a property.
It is assumed to be a desirable property, and is enforced in the code and
unit tests.
\end_layout
\begin_layout Itemize
Counts will be denoted with a lower-case n written in front of the pair.
Thus,
\family typewriter
n(A, BC)
\family default
would be the number of times that
\family typewriter
(A, BC)
\family default
was observed.
\end_layout
\begin_layout Subsubsection*
First merge
\end_layout
\begin_layout Standard
The troublesome test is '
\family typewriter
connector-merge-tricon.scm
\family default
'.
The relevant portion is as follows.
The dictionary is assumed to contain many entries; the troublesome subset
is this:
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
(j, abe)
\end_layout
\begin_layout Plain Layout
(f, abe)
\end_layout
\end_inset
A decision is made to merge the vectors for
\family typewriter
e
\family default
and
\family typewriter
j
\family default
, based on other dictionary entries not shown here.
The
\begin_inset Quotes eld
\end_inset
projective merge
\begin_inset Quotes erd
\end_inset
strategy is used, so that a fraction
\begin_inset Formula $0\le p\le1$
\end_inset
of the count is merged whenever one of the two vectors is missing an entry
at a given basis element.
In this case, the merge, denoted with an arrow, is
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
none + (j, abe) -> p * ({ej}, abe) + (1-p) * (j, abe)
\end_layout
\end_inset
where '
\family typewriter
none
\family default
' denotes that there is no section
\family typewriter
(e, abe)
\family default
and so the projective merge was used.
That is, the count on
\family typewriter
(j, abe)
\family default
is reduced to
\begin_inset Formula $\left(1-p\right)$
\end_inset
of it's earlier value, and the remaining
\begin_inset Formula $p$
\end_inset
is transferred over to
\family typewriter
({ej}, abe)
\family default
.
That is, the total counts are preserved.
That is,
\begin_inset Formula
\[
n^{\prime}\left(\texttt{\{ej\}, abe}\right)=pn\left(\texttt{j, abe}\right)
\]
\end_inset
where
\begin_inset Formula $n^{\prime}$
\end_inset
denotes the count after the merge, and the unprimed
\begin_inset Formula $n$
\end_inset
is the count before the merge.
\end_layout
\begin_layout Standard
From the connector merging discussion above, we conclude that
\family typewriter
({ej}, abe)
\family default
should be rewritten to
\family typewriter
({ej}, ab{ej})
\family default
.
The count should be as above, that is:
\begin_inset Formula
\[
n^{\prime}\left(\texttt{\{ej\}, ab\{ej\}}\right)=pn\left(\texttt{j, abe}\right)
\]
\end_inset
\end_layout
\begin_layout Standard
The vector on
\family typewriter
e
\family default
includes the cross-sections
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
[e, <j,abv>]
\end_layout
\begin_layout Plain Layout
[e, <f,abv>]
\end_layout
\end_inset
These merge in a similar fashion:
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
[e, <j,abv>] + none -> p * [{ej}, <j,abv>] + (1-p) * [e, <j,abv>]
\end_layout
\begin_layout Plain Layout
[e, <f,abv>] + none -> p * [{ej}, <f,abv>] + (1-p) * [e, <f,abv>]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
From detailed balance, we deduce the two new sections
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
(j, ab{ej})
\end_layout
\begin_layout Plain Layout
(f, ab{ej})
\end_layout
\end_inset
\end_layout
\begin_layout Standard
The counts on these are
\begin_inset Formula
\[
n^{\prime}\left(\texttt{j, ab\{ej\}}\right)=pn\left(\texttt{j, abe}\right)
\]
\end_inset
and
\begin_inset Formula
\[
n^{\prime}\left(\texttt{f, ab\{ej\}}\right)=pn\left(\texttt{f, abe}\right)
\]
\end_inset
From the connector merging discussion above, we conclude that
\family typewriter
(j, ab{ej})
\family default
should be rewritten to
\family typewriter
({ej}, ab{ej})
\family default
.
The first identity arrives at the same count as before, so this rewrite
appears to be self-consistent.
Everything works out.
\end_layout
\begin_layout Subsubsection*
Second merge
\end_layout
\begin_layout Standard
The first merge is more-or-less straightforward.
The trouble comes with the second merge.
Here, it is decided that the vector for
\family typewriter
f
\family default
should be merged into
\family typewriter
{ej}
\family default
.
The preservation of detailed balance creates subtleties and ambiguities.
\end_layout
\begin_layout Standard
The final count on
\family typewriter
({ejf}, ab{ejf})
\family default
gets contributions from three sources:
\end_layout
\begin_layout Itemize
The starting count on
\family typewriter
({ej}, ab{ej})
\family default
, which is
\begin_inset Formula
\[
n^{\prime}\left(\texttt{\{ej\}, ab\{ej\}}\right)=pn\left(\texttt{j, abe}\right)
\]
\end_inset
as given above.
\end_layout
\begin_layout Itemize
A contribution from
\family typewriter
(f, ab{ej})
\family default
, which was created in the first merge, via detailed balance from the earlier
cross-section
\family typewriter
[{ej}, <f, abv>]
\family default
.
This is merged in it's entirety into the the existing
\family typewriter
({ej}, ab{ej})
\family default
.
The projection merge is
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
({ej}, ab{ej}) + (f, ab{ej}) -> ({ejf}, ab{ej})
\end_layout
\end_inset
This merge absorbs the entire count on
\family typewriter
(f, ab{ej})
\family default
because
\family typewriter
({ej}, ab{ej})
\family default
already exists.
The contribution is thus
\begin_inset Formula $n^{\prime}\left(\texttt{f, ab\{ej\}}\right)=pn\left(\texttt{f, abe}\right)$
\end_inset
.
Rewriting then promotes
\family typewriter
({ejf}, ab{ej})
\family default
to
\family typewriter
({ejf}, ab{ejf})
\family default
.
\end_layout
\begin_layout Itemize
A contribution from
\family typewriter
(f, abe)
\family default
via the projection merge
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
none + (f, abe) -> q * ({ejf}, abe) + (1-q) * (f, abe)
\end_layout
\end_inset
and then the subsequent rewrite of
\family typewriter
({ejf}, abe) -> ({ejf}, ab{ejf})
\family default
.
This contribution is
\begin_inset Formula $qn^{\prime}\left(\texttt{f, abe}\right)=q\left(1-p\right)n\left(\texttt{f, abe}\right)$
\end_inset
.
\end_layout
\begin_layout Standard
The total of these three contributions is then
\begin_inset Formula
\begin{align*}
n^{\prime\prime}\left(\texttt{\{ejf\},ab\{ejf\}}\right)= & n^{\prime}\left(\texttt{\{ej\}, ab\{ej\}}\right)+n^{\prime}\left(\texttt{f, ab\{ej\}}\right)+qn^{\prime}\left(\texttt{f, abe}\right)\\
= & pn\left(\texttt{j, abe}\right)+pn\left(\texttt{f, abe}\right)+q\left(1-p\right)n\left(\texttt{f, abe}\right)
\end{align*}
\end_inset
Note that this result is history-dependent: merging j into e first, then
f gives a different result than merging f into e, then j (and presumably
different than the third possibility, of merging f and j first, and only
then adding e).
\end_layout
\begin_layout Standard
An open question is whether there is a way of performing the merges that
are history-independent, and what would that mean.
\end_layout
\begin_layout Standard
Again, additional details are in the test file '
\family typewriter
connector-merge-tricon.scm
\family default
'.
\end_layout
\begin_layout Subsection*
Connector Merging, Conclusion
\end_layout
\begin_layout Standard
After much work: there are ten unit tests, all passing, with the final fix
in commit 5e1d7dfb94867f22642d7cdf0621a833bb96092e of 24 May 2021 which
fixes a problem not found in the unit tests; it requires a real-world test-case.
Need to run
\family typewriter
(check-balance LLOBJ)
\family default
to evoke it.
\end_layout
\begin_layout Subsection*
expt-19 (May 2021)
\end_layout
\begin_layout Standard
Moving on...
expt-19 reuses the same corpus as expt-16, and, in order to be comparable
to earlier results, reuses the
\family typewriter
pair-counts
\family default
and the
\family typewriter
mpg-parse
\family default
disjuncts from expt-16.
Here's the dataset statistics, from
\family typewriter
print-matrix-summary-report
\family default
, without and with shapes:
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="9" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
cset-only
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
w/ shapes
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
rows
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
12
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
12
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
columns
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
75688
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
587172
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
entries
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
80832
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
701606
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
sparsity
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.4901
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.3281
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
avg.
obs./disjunct
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
12.561
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
6.4922
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
entropy
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.7221
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
5.0270
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
MMT support
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
80832
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
701606
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
obs.
count
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.9e9
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
15e9
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Standard
Time to compute the pair-distances was about 0.03 to 1.5 seconds for the cset-only
vectors, and 6 to 16 seconds for the shapes.
So, at least an order of magnitude slower.
That's bad.
\end_layout
\begin_layout Standard
The entropy is much much higher, which I interpret as a good thing, indicating
that the data is of higher quality.
\end_layout
\begin_layout Standard
Using
\family typewriter
(gram-classify-greedy-disinfo psa 3.0 4)
\family default
, there weren't any merges that got done.
That's because similarity never got above 3.0.
Here's a table of all positive MI similarities, without shapes, and with.
It's sorted on the shape column when the shape column MI is positive, otherwise
sorted on the cset-only column.
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="25" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
pair
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
MI cset-only
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
MI w/ shapes
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
!-i
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $-\infty$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.3578
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
g-i
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.2143
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.8643
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
a-i
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-2.388
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.2513
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
f-c
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.3032
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.1714
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
h-c
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.2824
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.0837
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
f-h
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.2800
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.0757
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
b-g
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.8982
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.0559
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
a-WALL
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $-\infty$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.9799
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
j-b
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.3929
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.5846
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
j-g
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.3929
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.5265
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
d-f
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.8141
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.5170
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
e-b
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.3725
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.5066
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
d-c
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.7817
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.5026
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
e-j
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.2484
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.4612
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
e-g
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.3896
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.4503
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
d-h
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.8126
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.4446
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
e-d
\end_layout
\end_inset
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\end_layout
\begin_layout Standard
In general, any MI of less than 4 is ...
pretty distant.
Although that statement is true for large-vocabulary systems; here the
vocabulary is tiny: 12 words, so I guess an MI greater than 1.0 is actually
pretty good.
\end_layout
\begin_layout Standard
The distance `!-i` is alarming.
I guess.
Seems to indicate that `i` is usually at the end of sentences.
The `a-WALL` distance suggests that `a` is frequently at the start of the
sentence.
If so, this is not obvious from casual inspection of the corpus.
\end_layout
\begin_layout Standard
A core problem is that this is a very mixed grammar: lots of ambiguity,
lots of word senses, no particular clean factorization.
Just looking at it, its quite cloudy as to the actual structure.
In particular, although each word belongs to only one word-class, the word-clas
ses have lots of mixed, shared POS entries, and each POS is a seemingly
random (duhh) unstructured mess.
For example:
\end_layout
\begin_layout Itemize
pos-e has single, divalent, trivalent disjuncts on it.
Except for a few words, most of English is not like that.
\end_layout
\begin_layout Itemize
pos-c has one disjunct.
It's identical to pos-d, which has two disjuncts.
\end_layout
\begin_layout Itemize
pos-e has a huge number of disjuncts on it, as do pos-i and pos-j.
This seems to allow very grammatically complex sentences to be generated,
which
\begin_inset Quotes eld
\end_inset
of course
\begin_inset Quotes erd
\end_inset
are going to be very hard to decode.
\end_layout
\begin_layout Standard
Overall, the grammar appears to be over-complex, and very unlike a natural
language grammar.
it seems unlikely, just from eyeballing it, that the grammar could be untangled
without a very large, exhaustive examination of the corpus.
This is a bad experimental base.
\end_layout
\begin_layout Standard
Issues:
\end_layout
\begin_layout Itemize
Is there any sense in which the word
\begin_inset Quotes eld
\end_inset
mixing
\begin_inset Quotes erd
\end_inset
is appropriate, in it's technical sense (from ergodic theory?) Can we define
mixing from the point of view of disjunct ambiguity? Of the indiscernability
of grammars given a corpus?
\end_layout
\begin_layout Itemize
Is the automatic grammar generation controlling sufficiently for word-senses?
Yes, there's a tune-able parameter for that, but some disjuncts accidentally
appear in multiple POS, thus making those POS at least partly synonymous.
\end_layout
\begin_layout Itemize
Is the current automatic grammar generation API appropriate? It was based
on an intuitive sense of factorization, but the randomness seems to easily
generate ambiguous grammars.
\end_layout
\begin_layout Itemize
How does one measure the complexity of a grammar?
\end_layout
\begin_layout Itemize
Is there some easy way of writing down its factorizability?
\end_layout
\begin_layout Itemize
Is there a way of proving that two grammars are equivalent? If two different
grammars generate the exact same corpus, then is there some algorithm that
can transmute one grammar into the other? How is this found/discovered?
\end_layout
\begin_layout Itemize
How can one characterize human natural language grammars? That is, if an
artificial grammar is generated, how can we know if it is similar to a
natural human language? I don't think the rainbow of human natural languages
lines up well (or at all) with the axes of tune-able parameters in the
grammar generator.
\end_layout
\begin_layout Standard
Conclude: the expt-19 grammar is over-complicated, ambiguous, mixed, ugly.
We need to restart with a simple grammar.
\end_layout
\begin_layout Standard
Also conclude: I do not understand how the ambiguity of grammars works.
I do not really understand how factorization works.
The artificial grammar generator
\begin_inset Quotes eld
\end_inset
works
\begin_inset Quotes erd
\end_inset
but I don't understand what it is generating.
I don't know how close it is to typical human grammars.
There's a bit of a
\begin_inset Quotes eld
\end_inset
back to the drawing board
\begin_inset Quotes erd
\end_inset
moment here.
\end_layout
\begin_layout Subsection*
expt-20 (May 2021)
\end_layout
\begin_layout Standard
Start again, this time with a simple, relatively unambiguous, relatively
unmixed grammar.
Perhaps even with a tiny artificial subset of English!? Just to make eyeballing
easier?
\end_layout
\begin_layout Section*
July 2021 - Projective Merge and Entropy Maximization
\end_layout
\begin_layout Standard
OK, Above work resulted in a bunch of automation scripts, and a few bug-fixes,
a completion of the
\begin_inset Quotes eld
\end_inset
shapes
\begin_inset Quotes erd
\end_inset
work, but not much in the way of insight.
So, restart processing of English.
A rather small corpus reveals a conceptual bug in the projective-merge
strategy, when used with mutual information.
Projective merge works great (I think ..
I guess??) with cosine distance, but not MI.
This section describes the problem, and explores solutions.
\end_layout
\begin_layout Standard
The small corpus is the
\begin_inset Quotes eld
\end_inset
run-2
\begin_inset Quotes erd
\end_inset
corpus of the English work, its a truncated copy of
\begin_inset Quotes eld
\end_inset
tranche-1
\begin_inset Quotes erd
\end_inset
consisting of 3026 articles, 426941 sentences, 8133834 word instances.
Word-pair counting went well (except for the handling error that truncated
the corpus), as did disjunct formation.
The MM^T stats were computed for the disjuncts, including shapes.
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
It contains 6029 words, 98102 disjuncts, a total of 171922 word-disjunct
pairs, has a sparsity of 11.748, a total of 4006152 observations for an
average of 23.302 per pair.
\end_layout
\end_inset
Everything is fine (nominal) until merge.
Projective merge fails dramatically – unrelated words are being merged.
Why?
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
One possible answer to this question is that there is a procedural error
at play.
It is this: the MM^T marginals were computed for the full dataset, and
then the dataset was trimmed to discard low-count rows and columns.
However, the full marginals were still used in computing the MI.
This throws everything off, because ..
rows and columns are missing.
The correct process would have been to recompute the MM^T marginals
\emph on
after
\emph default
trimming! The MM^T marginals depend on the support marginals as well.
Which were also cached.
This has lead to some confusion in the below.
I think the results are unchanged, but I'm no longer confident.
27 July 2021
\end_layout
\end_inset
\end_layout
\begin_layout Standard
What's the pair-wise MI? The MI here is called the
\begin_inset Quotes eld
\end_inset
entropic similarity
\begin_inset Quotes erd
\end_inset
in other texts in this directory.
It's kind-of the same thing, once one adjusts definitions appropriately.
Lets call it MI for short.
Lets explore the top seven words that got merged, shown in the table below.
\end_layout
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\begin_inset Text
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\begin_inset Text
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\begin_inset Text
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3.223
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</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
Looks pretty reasonable, right? The self-MI of
\begin_inset Quotes eld
\end_inset
it
\begin_inset Quotes erd
\end_inset
is shockingly low.
Some of the other self-MI's are on the low side, too.
But whatever.
Small dataset.
Asking for an MI > 3.0 for a merge to take place seems like it should provide
good results.
\end_layout
\begin_layout Standard
The current code base for projective merge recommends a fraction of 0.187
for merging
\begin_inset Quotes eld
\end_inset
of
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
to
\begin_inset Quotes erd
\end_inset
, which seems low, but acceptable.
So lets do the merge, by hand (see notes in `run-2/README` for details.
The result is `(WordClassNode
\begin_inset Quotes eld
\end_inset
of to
\begin_inset Quotes erd
\end_inset
)`.
What is the MI between this, and the other words? A disaster.
See table below.
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="9">
<features tabularvalignment="middle">
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<column alignment="center" valignment="top">
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\begin_inset Text
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Symmetric-MI
\end_layout
\end_inset
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\end_layout
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\begin_inset Text
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of-to
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of
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to
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in
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he
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it
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that
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said
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\begin_inset Text
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10.009
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8.411
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8.216
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6.044
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2.970
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of
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11.70
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\begin_inset Formula $-\infty$
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5.294
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-4.44
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-3.34
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1.012
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</cell>
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\begin_inset Text
\begin_layout Plain Layout
to
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\begin_inset Text
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\begin_inset Text
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11.92
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\begin_inset Text
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3.558
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0.874
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1.432
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-1.97
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</cell>
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</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
OK, so what's wrong with that? The naive expectation is that the MI(cluster,
\begin_inset Quotes eld
\end_inset
of
\begin_inset Quotes erd
\end_inset
) and MI(cluster,
\begin_inset Quotes eld
\end_inset
to
\begin_inset Quotes erd
\end_inset
) should be low or negative.
It's not - its huge.
The naive expectation is that the MI of the cluster to the non-preposition
words should remain near zero (small positive or negative) and instead
it got higher, not lower! Yow!
\end_layout
\begin_layout Standard
Note that the
\begin_inset Quotes eld
\end_inset
of
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
to
\begin_inset Quotes erd
\end_inset
vectors are the new vectors, with the projected parts removed.
Thus, the new MI (of,to)=
\begin_inset Formula $-\infty$
\end_inset
is as expected: all overlap is now completely gone.
That the MI(of, non-prep) and MI(to, non-prep) has gotten lower is a good
sign.
\end_layout
\begin_layout Standard
So what's going wrong, here? Well, we failed to define the projective merge
in such a way that we minimize and maximize the assorted overlaps.
So lets fix that.
\end_layout
\begin_layout Subsubsection*
Conclusion
\end_layout
\begin_layout Standard
The following sections explore the problem above.
The answer, in retrospect, appears to be obvious: of the parameterized
projective merge, only the overlap merge maximizes entropy.
Adding in any fraction of the union merge lowers the total mutual information.
It smears out the word-senses, and damages word-sense disambiguation.
\end_layout
\begin_layout Standard
If the answer is so obvious, then how did we get into this mess to begin
with? Well, it seemed, at the time, that, because the input data is so
noisy, that the number of observations is so incomplete, that there might
be some advantage for extending the overlap merge with some
\begin_inset Quotes eld
\end_inset
small
\begin_inset Quotes erd
\end_inset
fraction of the union merge.
In retrospect, this appears to have been a failed idea.
\end_layout
\begin_layout Standard
The overall concern is still valid: for a small corpus, there is a lot of
noise, and perhaps there does need to be some kind of generalization taking
place.
But perhaps the smearing provided by the union merge is ...
not a good idea ...
at present.
However ...
\end_layout
\begin_layout Standard
However, trying to always maximize the entropy runs the risk of stumbling
into local maxima, and being trapped by them.
This is an old machine-learning problem: how to avoid local maxima.
The union merge might eventually be a good way of jumping out of local
maxima.
However, at this time, I don't understand what's going on clearly enough.
It seems to have been a premature feature.
Maybe.
At any rate, future work should set a baseline of pure overlap merges,
zero union merges.
If any part of the union is to be mixed in, it should probably be quite
small – well less than a percent, rather than the 30% which almost all
earlier work made use of.
\end_layout
\begin_layout Standard
If the union-fraction is greater than zero, then it should be kept small
enough so that the MI between the newly created cluster, and the remainder
of the words that were put into the cluster remains less than the cutoff
MI that is used to determine if a merge is to be undertaken.
Examining the experimental data, below, indicates that this fraction is
indeed tiny: its about 0.003 (a third of a percent) for the case that was
examined.
\end_layout
\begin_layout Standard
Rule of thumb: if there is a merge fraction, then if should be set to
\begin_inset Formula
\[
f=1/2^{\max(MI(a,a),MI(b,b))-MI_{cut}}
\]
\end_inset
where
\begin_inset Formula $a,b$
\end_inset
are the two words to be merged,
\begin_inset Formula $MI(a,a)$
\end_inset
is the self-MI and
\begin_inset Formula $MI_{cut}$
\end_inset
is the cutoff, below which a merge will not be performed.
Any fraction larger than this will results in an MI between the new class,
and the remainder of the words
\begin_inset Formula $a,b$
\end_inset
that is
\emph on
larger
\emph default
than the original
\begin_inset Formula $MI(a,b)$
\end_inset
itself ...
and that's a disaster.
This is a rule of thumb, because, I guess a more precise value could be
given, but getting it requires some more algebra which I'm too lazy to
do.
\end_layout
\begin_layout Subsection*
Similarity and Projection
\end_layout
\begin_layout Standard
Let's review the definition for the MI, and the projection.
\end_layout
\begin_layout Subsubsection*
Entropic Similarity and Cosine Distance
\end_layout
\begin_layout Standard
This is a copy of what is in the `connector-sets-revised.pdf` paper, chapter
6 (pages 39-44).
There some additional experimental data in the `diary-part-one.pdf`, pages
102-103.
\end_layout
\begin_layout Standard
A word
\begin_inset Formula $w$
\end_inset
is associated with a vector
\begin_inset Formula $N\left(w,d_{j}\right)$
\end_inset
where the
\begin_inset Formula $d_{j}$
\end_inset
are the disjuncts observed on the word, and
\begin_inset Formula $N$
\end_inset
is a count of the number of times that the word-disjunct pair was observed.
\end_layout
\begin_layout Standard
For words
\begin_inset Formula $w,u$
\end_inset
, define the dot product (inner product) between the words as
\begin_inset Formula
\[
i\left(u,w\right)=\sum_{d}N\left(u,d\right)N\left(w,d\right)
\]
\end_inset
This can be turned into a
\emph on
bona-fide
\emph default
joint probability by writing
\begin_inset Formula
\[
p\left(u,w\right)=i\left(u,w\right)/i\left(*,*\right)
\]
\end_inset
where, as always, * denotes a wild-card – here, a sum over all words:
\begin_inset Formula
\[
i\left(*,*\right)=\sum_{u,w}i\left(u,w\right)
\]
\end_inset
\end_layout
\begin_layout Standard
There is a corresponding marginal probability
\begin_inset Formula
\[
p\left(w\right)=p\left(w,*\right)=\frac{i\left(w,*\right)}{i\left(*,*\right)}
\]
\end_inset
\end_layout
\begin_layout Standard
The entropic similarity between two words is
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
MI\left(u,w\right)=\log_{2}\frac{p\left(u,w\right)}{p\left(u\right)p\left(w\right)}
\]
\end_inset
and written in this form, this is clearly the conventional (fractional)
MI between two words.
\end_layout
\begin_layout Standard
It's worth comparing this to the cosine distance
\begin_inset Formula
\[
\theta\left(u,w\right)=\arccos\frac{p\left(u,w\right)}{\sqrt{p\left(u,u\right)p\left(w,w\right)}}
\]
\end_inset
Note that they both start with the dot product in the numerator.
The cosine distance is invariant under rotations (orthogonal transformations)
in Euclidean space.
However, probability space is not Euclidean, so it is a
\begin_inset Quotes eld
\end_inset
category error
\begin_inset Quotes erd
\end_inset
to work with cosine distance applied to probabilities.
By contrast, the MI is
\begin_inset Quotes eld
\end_inset
obviously correct
\begin_inset Quotes erd
\end_inset
when working with probabilities.
In practice, the two are correlated.
See page 43 of `connector-sets-revised.pdf` for experimental data.
\end_layout
\begin_layout Standard
The code for computing this stuff can be found in the AtomSpace github repo,
https://github.com/opencog/atomspace/blob/master/opencog/matrix/symmetric-mi.scm
To avoid absurdly long compute times, this code uses the MM^T concept (describe
d in connector-sets) to perform and cache partial results that can be quickly
combined to obtain the desired MI value.
Note, its impossible to pre-compute the MI for any but the smallest vocabularie
s, as there are just too many words.
\end_layout
\begin_layout Subsubsection*
Projection Merge
\end_layout
\begin_layout Standard
Given two words (word-vectors), the projection merge create three new vectors:
a merged vector, and remainders of the two original vectors.
This is most easily described in terms of the basis vectors.
Define the set of all disjuncts with non-zero counts on word
\begin_inset Formula $w$
\end_inset
:
\begin_inset Formula
\[
D\left(w\right)=\left\{ d:N\left(w,d\right)\ne0\right\}
\]
\end_inset
Given a real number
\begin_inset Formula $0\le f\le1$
\end_inset
, the merged vector
\begin_inset Formula $g$
\end_inset
of the two words
\begin_inset Formula $w,u$
\end_inset
has counts
\begin_inset Formula
\[
N\left(g,d\right)=\begin{cases}
N\left(w,d\right)+N\left(u,d\right) & d\in D\left(w\right)\cap D\left(u\right)\\
fN\left(w,d\right) & d\in D\left(w\right)\backslash D\left(u\right)\\
fN\left(u,d\right) & d\in D\left(u\right)\backslash D\left(w\right)
\end{cases}
\]
\end_inset
Clearly, the support for
\begin_inset Formula $g$
\end_inset
is
\begin_inset Formula $D\left(g\right)=D\left(w\right)\cup D\left(u\right)$
\end_inset
whenever
\begin_inset Formula $f>0$
\end_inset
and it is
\begin_inset Formula $D\left(g\right)=D\left(w\right)\cap D\left(u\right)$
\end_inset
when
\begin_inset Formula $f=0$
\end_inset
.
The two new words are
\begin_inset Formula $u^{\prime}$
\end_inset
and
\begin_inset Formula $w^{\prime}$
\end_inset
which are just the old words, with the overlaps removed:
\begin_inset Formula
\[
N\left(w^{\prime},d\right)=\begin{cases}
0 & d\in D\left(w\right)\cap D\left(u\right)\\
\left(1-f\right)N\left(w,d\right) & d\in D\left(w\right)\backslash D\left(u\right)
\end{cases}
\]
\end_inset
and likewise for
\begin_inset Formula $N\left(u^{\prime},d\right)$
\end_inset
.
\end_layout
\begin_layout Standard
This projection merge is designed to preserve the total count:
\begin_inset Formula
\begin{equation}
N\left(g,d\right)+N\left(w^{\prime},d\right)+N\left(u^{\prime},d\right)=N\left(w,d\right)+N\left(u,d\right)\label{eq:detailed balance}
\end{equation}
\end_inset
This equation can be called
\begin_inset Quotes eld
\end_inset
detailed balance
\begin_inset Quotes erd
\end_inset
, as it is in thermodynamics.
\end_layout
\begin_layout Standard
The merged vector
\begin_inset Formula $g$
\end_inset
is meant to correspond to a
\begin_inset Quotes eld
\end_inset
grammatical class
\begin_inset Quotes erd
\end_inset
; it is a vector just as any other word-vector, but it is meant to capture
the
\begin_inset Quotes eld
\end_inset
average
\begin_inset Quotes erd
\end_inset
of the two words it is made out of.
The role of the fraction
\begin_inset Formula $f$
\end_inset
is to handle the situation of the words
\begin_inset Formula $u$
\end_inset
and
\begin_inset Formula $w$
\end_inset
having multiple word-senses.
The idea here is that the set
\begin_inset Formula $D\left(w\right)\cap D\left(u\right)$
\end_inset
captures those disjuncts for which both
\begin_inset Formula $u$
\end_inset
and
\begin_inset Formula $w$
\end_inset
have the same word-sense (for example, the parts of
\begin_inset Formula $u$
\end_inset
and
\begin_inset Formula $w$
\end_inset
that are nouns) whereas
\begin_inset Formula $D\left(w\right)\backslash D\left(u\right)$
\end_inset
and
\begin_inset Formula $D\left(u\right)\backslash D\left(w\right)$
\end_inset
are the disjuncts that belong to other word-senses (
\emph on
e.g.
\emph default
the parts of
\begin_inset Formula $u$
\end_inset
and
\begin_inset Formula $w$
\end_inset
that are verbs).
Due to inadequate statistics and systemic noise, a non-zero
\begin_inset Formula $f$
\end_inset
might help in smoothing out erroneous assignments of disjuncts to word-senses.
In the example above, a value of
\begin_inset Formula $f=0.187$
\end_inset
was used.
\end_layout
\begin_layout Standard
The projection described above is the same as that described and implemented
in https://github.com/opencog/learn/blob/master/scm/gram-projective.scm (as
of this writing).
\end_layout
\begin_layout Subsection*
Maximum Mutual Information
\end_layout
\begin_layout Standard
Well, obviously as the example above demonstrates, there's something not
quite right with the projection merge, at least, when one works with mutual
information.
\end_layout
\begin_layout Standard
The projection merge does seem to make sense (mostly) when one thinks of
vectors that inhabit Euclidean space: One can easily define the sum of
two vectors; and to deal with the fact that the word vectors are themselves
sums of multiple senses, one can try to project back out the parts that
don't have shared support.
The problem with working in Euclidean space is that the counts can go negative:
the inner product on Euclidean space really is the cosine (dot) product,
and simple linear algebra sense some vector components negative.
This is undesirable, as it prevents counts from being interpreted as statistica
l frequencies.
\end_layout
\begin_layout Standard
For probabilities, the correct goal is to maximize the mutual information
(maximize the entropy).
There seem to be two ways to define the total MI of the system.
One is to write
\begin_inset Formula
\[
MI_{tot}=\sum_{u,w}p\left(u,w\right)MI\left(u,w\right)
\]
\end_inset
Note that since both
\begin_inset Formula $p$
\end_inset
and MI are symmetric, this has the effect of double-counting the off-diagonal
entries.
This seems to underweight the diagonal.
Another possibility is to double-weight the diagonal:
\begin_inset Formula
\[
MI_{alt}=\sum_{u\ge w}p\left(u,w\right)MI\left(u,w\right)
\]
\end_inset
Not clear, right now, which is better or more correct.
\end_layout
\begin_layout Standard
A bit of articulation helps clarify things.
From the definition of MI, one has
\begin_inset Formula
\begin{align*}
MI_{tot}= & \sum_{u,w}p\left(u,w\right)MI\left(u,w\right)\\
= & \sum_{u,w}p\left(u,w\right)\log_{2}\frac{p\left(u,w\right)}{p\left(u\right)p\left(w\right)}\\
= & \sum_{u,w}p\left(u,w\right)\log_{2}p\left(u,w\right)-\sum_{u,w}p\left(u,w\right)\log_{2}p\left(u\right)-\sum_{u,w}p\left(u,w\right)\log_{2}p\left(w\right)\\
= & \sum_{u,w}p\left(u,w\right)\log_{2}p\left(u,w\right)-2\sum_{w}p\left(w\right)\log_{2}p\left(w\right)\\
= & H_{joint}-2H_{marg}
\end{align*}
\end_inset
where
\begin_inset Formula
\[
H_{marg}=\sum_{w}p\left(w\right)\log_{2}p\left(w\right)
\]
\end_inset
is the
\begin_inset Quotes eld
\end_inset
marginal entropy
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Subsubsection*
Detailed balance
\end_layout
\begin_layout Standard
Detailed balance, eqn
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:detailed balance"
plural "false"
caps "false"
noprefix "false"
\end_inset
means that the merge affects only the rows and columns of the merged words.
This is somehow
\begin_inset Quotes eld
\end_inset
intuitively obvious
\begin_inset Quotes erd
\end_inset
, but I'll belabor the topic here, to make sure we're not making any mistakes.
In the following, let
\begin_inset Formula $a,b$
\end_inset
be the two words to be merged, creating a category
\begin_inset Formula $g$
\end_inset
.
\end_layout
\begin_layout Standard
\series bold
Lemma:
\series default
For
\begin_inset Formula $w\ne a,b$
\end_inset
, the row and column sums are unaffected by the merge.
That is, one has
\begin_inset Formula $i\left(w,*\right)=i^{\prime}\left(w,*\right)$
\end_inset
.
\end_layout
\begin_layout Standard
\emph on
Proof:
\emph default
Let
\begin_inset Formula $a,b$
\end_inset
be the two words to be merged.
Then for
\begin_inset Formula $w\ne a,b$
\end_inset
, one has
\begin_inset Formula
\begin{align*}
i\left(w,a\right)+i\left(w,b\right)= & \sum_{d}N\left(w,d\right)\left[N\left(a,d\right)+N\left(b,d\right)\right]\\
= & \sum_{d}N\left(w,d\right)\left[N\left(g,d\right)+N\left(a^{\prime},d\right)+N\left(b^{\prime},d\right)\right]\\
= & i\left(w,g\right)+i\left(w,a^{\prime}\right)+i\left(w,b^{\prime}\right)
\end{align*}
\end_inset
No other sums are affected, so that, for
\begin_inset Formula $u,w\ne a,b$
\end_inset
one has
\begin_inset Formula
\[
i\left(w,u\right)=i^{\prime}\left(w,u\right)
\]
\end_inset
Therefore,
\begin_inset Formula
\begin{align*}
i\left(w,*\right)= & \sum_{u}i\left(w,u\right)\\
= & i\left(w,a\right)+i\left(w,b\right)+\sum_{u\ne a,b}i\left(w,u\right)\\
= & i\left(w,g\right)+i\left(w,a^{\prime}\right)+i\left(w,b^{\prime}\right)+\sum_{u\ne a,b}i\left(w,u\right)\\
= & i^{\prime}\left(w,*\right)
\end{align*}
\end_inset
where the
\begin_inset Formula $i^{\prime}$
\end_inset
sum includes (runs over)
\begin_inset Formula $a^{\prime},b^{\prime}$
\end_inset
and
\begin_inset Formula $g$
\end_inset
.
\begin_inset Formula $\square$
\end_inset
\end_layout
\begin_layout Standard
This enables a key theorem.
\end_layout
\begin_layout Standard
\series bold
Theorem:
\series default
The total count
\begin_inset Formula $i\left(*,*\right)$
\end_inset
is unaffected by the merge.
That is,
\begin_inset Formula $i^{\prime}\left(*,*\right)=i\left(*,*\right)$
\end_inset
where the prime denotes the post-merge sum.
\end_layout
\begin_layout Standard
\emph on
Proof:
\emph default
Split out the affected rows and columns.
First, the corner case:
\begin_inset Formula
\begin{align*}
i\left(a,a\right)+i\left(a,b\right)+i\left(b,a\right)+i\left(b,b\right)= & \sum_{d}\left[N\left(a,d\right)+N\left(b,d\right)\right]\left[N\left(a,d\right)+N\left(b,d\right)\right]\\
= & \sum_{d}\left[N\left(g,d\right)+N\left(a^{\prime},d\right)+N\left(b^{\prime},d\right)\right]\left[N\left(g,d\right)+N\left(a^{\prime},d\right)+N\left(b^{\prime},d\right)\right]\\
= & i\left(g,g\right)+i\left(g,a^{\prime}\right)+i\left(g,b^{\prime}\right)+i\left(a^{\prime},g\right)+i\left(a^{\prime},a^{\prime}\right)+i\left(b^{\prime},g\right)+i\left(b^{\prime},b^{\prime}\right)\\
= & i\left(g,g\right)+i\left(a^{\prime},a^{\prime}\right)+i\left(b^{\prime},b^{\prime}\right)+2i\left(g,a^{\prime}\right)+2i\left(g,b^{\prime}\right)
\end{align*}
\end_inset
Two of the terms vanish:
\begin_inset Formula $i\left(a^{\prime},b^{\prime}\right)=i\left(b^{\prime},a^{\prime}\right)=0$
\end_inset
, which is arrived at by noting that the merge has arranged that
\begin_inset Formula $N\left(a^{\prime},d\right)=N\left(b^{\prime},d\right)=0$
\end_inset
for
\begin_inset Formula $d\in D\left(a\right)\cap D\left(b\right)$
\end_inset
.
\end_layout
\begin_layout Standard
Even if these terms did not vanish, one still arrives at
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
i\left(a,*\right)+i\left(b,*\right)=i\left(g,*\right)+i\left(a^{\prime},*\right)+i\left(b^{\prime},*\right)
\]
\end_inset
Plugging through,
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align*}
i\left(*,*\right)= & \sum_{u}i\left(u,*\right)\\
= & i\left(a,*\right)+i\left(b,*\right)+\sum_{u\ne a,b}i\left(u,*\right)\\
= & i\left(g,*\right)+i\left(a^{\prime},*\right)+i\left(b^{\prime},*\right)+\sum_{u\ne a,b}i\left(u,*\right)\\
= & \sum_{u^{\prime}}i\left(u^{\prime},*\right)\\
= & i^{\prime}\left(*,*\right)
\end{align*}
\end_inset
Phew.
That was complicated, given that the result seems obvious.
\begin_inset Formula $\square$
\end_inset
\end_layout
\begin_layout Standard
\series bold
Corollary
\series default
: Likewise, for the probabilities: if
\begin_inset Formula $a,b$
\end_inset
are the two words to be merged into
\begin_inset Formula $g,$
\end_inset
then
\begin_inset Formula $p\left(a,*\right)+p\left(b,*\right)=p\left(g,*\right)+p\left(a^{\prime},*\right)+p\left(b^{\prime},*\right)$
\end_inset
.
\end_layout
\begin_layout Standard
\emph on
Proof:
\emph default
Divide by
\begin_inset Formula $i\left(*,*\right)$
\end_inset
.
\begin_inset Formula $\square$
\end_inset
\end_layout
\begin_layout Standard
\series bold
Corollary:
\series default
The marginal probability of rows/columns not being merged is unaffected
by the merge.
That is,
\begin_inset Formula $p\left(w\right)=p^{\prime}\left(w\right)$
\end_inset
for
\begin_inset Formula $w\ne a,b$
\end_inset
.
\end_layout
\begin_layout Standard
\emph on
Proof:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
p\left(w\right)=p\left(w,*\right)=\frac{i\left(w,*\right)}{i\left(*,*\right)}=\frac{i^{\prime}\left(w,*\right)}{i^{\prime}\left(*,*\right)}=p^{\prime}\left(w\right)
\]
\end_inset
Follows from the lemma and the theorem above.
\begin_inset Formula $\square$
\end_inset
\end_layout
\begin_layout Standard
\series bold
Corollary:
\series default
The mutual information of rows/columns not being merged is unaffected by
the merge.
That is,
\begin_inset Formula $MI\left(u,w\right)=MI^{\prime}\left(u,w\right)$
\end_inset
for
\begin_inset Formula $u,w\ne a,b$
\end_inset
.
\end_layout
\begin_layout Standard
\emph on
Proof:
\emph default
\begin_inset Formula
\begin{align*}
MI\left(u,w\right)= & \log_{2}\frac{p\left(u,w\right)}{p\left(u\right)p\left(w\right)}\\
= & \log_{2}\frac{p^{\prime}\left(u,w\right)}{p^{\prime}\left(u\right)p^{\prime}\left(w\right)}=MI^{\prime}\left(u,w\right)
\end{align*}
\end_inset
Follows as before.
\begin_inset Formula $\square$
\end_inset
\end_layout
\begin_layout Standard
From the above, it is clear that the
\begin_inset Formula $MI_{tot}$
\end_inset
splits into an invariant part, and a part affected by the merge.
Write
\begin_inset Formula
\begin{align*}
MI_{tot}= & \frac{1}{2}\sum_{u,w}p\left(u,w\right)MI\left(u,w\right)\\
= & \frac{1}{2}\sum_{u,w=a,b}p\left(u,w\right)MI\left(u,w\right)+\frac{1}{2}\sum_{u,w\ne a,b}p\left(u,w\right)MI\left(u,w\right)\\
= & MI_{merge}+MI_{invariant}
\end{align*}
\end_inset
The focus is then on how
\begin_inset Formula $MI_{merge}$
\end_inset
changes.
The change in the mutual information due to merging is captured by the
difference, defined as
\begin_inset Formula
\begin{equation}
S=\sum_{u,w=g,a^{\prime},b^{\prime}}p\left(u,w\right)MI\left(u,w\right)-\sum_{u,w=a,b}p\left(u,w\right)MI\left(u,w\right)\label{eq:relative entropy}
\end{equation}
\end_inset
A suitable name for this would be the
\begin_inset Quotes eld
\end_inset
relative entropy
\begin_inset Quotes erd
\end_inset
, I guess.
Seems reasonable.
\end_layout
\begin_layout Subsection*
Experimental Exploration
\end_layout
\begin_layout Standard
The goal of later sections will be to find the extrema (maxima, minima)
of eqn
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:relative entropy"
plural "false"
caps "false"
noprefix "false"
\end_inset
by algebraic means: that is, to find a value for the parameter
\begin_inset Formula $f$
\end_inset
that yields the projective merge that maximizes the mutual information.
As it turns out, a whole lot of rather tedious algebra is required.
Thus, its time for an experimental interlude.
Given the dataset above, we can vary the merge parameter by hand, and see
what happens.
\end_layout
\begin_layout Standard
First, a review of the dataset and the computational techniques.
The dataset is `r2-mpg-trim-40-8-5.rdb`.
It contains 6029 words and 98102 disjuncts, and a total of 171922 word-disjunct
pairs that were observed a total of 4006152 times.
That is, each word/disjunct pair was observed an average of 23.302 times.
The dataset is very sparse, with a log2 sparsity of 11.748.
This is a trimmed dataset: all words with less than 40 observations were
discarded; all disjuncts with less than 8 observations were discarded,
and all pairs with less than 5 observations were discarded.
See the experiment README file for more info on row and column support,
average lengths, etc.
\end_layout
\begin_layout Standard
Here are the probabilities and MI, before merge:
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
word-pair
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $p\left(u,w\right)$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $MI\left(u,w\right)$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
of,of
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.199e-3
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.460
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
of,to
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.599e-5
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.720
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
to,to
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
5.623e-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
6.848
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
The total probability is then 1.913e-3 and the weighted MI is 1.336e-2 and
the average MI is then 6.9829.
To repeat:
\begin_inset Formula
\begin{align*}
\sum_{u,w=a,b}p\left(u,w\right) & =1.913\times10^{-3}\\
\sum_{u,w=a,b}p\left(u,w\right)MI\left(u,w\right) & =1.336\times10^{-2}\\
\frac{\sum_{u,w=a,b}p\left(u,w\right)MI\left(u,w\right)}{\sum_{u,w=a,b}p\left(u,w\right)} & =6.9829
\end{align*}
\end_inset
for
\begin_inset Formula $a,b=of,to$
\end_inset
.
\end_layout
\begin_layout Standard
Lets repeat this calculation, after merging, with
\begin_inset Formula $f=0.18715$
\end_inset
as before.
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="7" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
pair
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $p\left(u,w\right)$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $MI\left(u,w\right)$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
of,of
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4.831E-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
11.695
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
to,to
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.153E-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
11.923
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
g,g
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.465E-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
10.009
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
g,of
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.105E-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
8.4107
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
g,to
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.208E-5
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
8.2156
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
total
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.910E-3
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
10.431
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
The MI reported in the total column is the parameter-dependent entropy from
above:
\begin_inset Formula
\[
S^{\prime}=\sum_{u,w=g,a^{\prime},b^{\prime}}p\left(u,w\right)MI\left(u,w\right)
\]
\end_inset
where
\begin_inset Formula $a^{\prime}$
\end_inset
is what's left of the vector for the word
\begin_inset Quotes eld
\end_inset
of
\begin_inset Quotes erd
\end_inset
, after projection-merging the common part.
Likewise,
\begin_inset Formula $b^{\prime}$
\end_inset
is what's left of
\begin_inset Quotes eld
\end_inset
to
\begin_inset Quotes erd
\end_inset
.
It is numerically confirmed that
\begin_inset Formula $a^{\prime}$
\end_inset
and
\begin_inset Formula $b^{\prime}$
\end_inset
are orthogonal to each-other: that
\begin_inset Formula $p\left(a^{\prime},b^{\prime}\right)=0$
\end_inset
after the merge.
This is as expected — the common components have been merged.
That
\begin_inset Formula $p\left(g,a^{\prime}\right)\ne0$
\end_inset
is not surprising: an additional portion of
\begin_inset Formula $a$
\end_inset
was merged into
\begin_inset Formula $g$
\end_inset
, beyond what is strictly possible with the overlap merge.
That is,
\begin_inset Formula $g$
\end_inset
and
\begin_inset Formula $a^{\prime}$
\end_inset
are intentionally not orthogonal.
That's what this is all about – finding out if something broader than a
pure overlap merge is somehow advantageous.
\end_layout
\begin_layout Standard
Annoyingly, the detailed balance seems to be a bit off — 1.910e-3
\emph on
vs.
\emph default
1.913e-3 before.
That's a fairly hefty amount of rounding error.
Surprisingly large...
There are 7620 sections and 7620 shapes being merged; calculations should
be double-precision, so this rounding error is irritating.
\end_layout
\begin_layout Standard
Anyway, that's the baseline.
Lets try a range of merge fractions.
Again, to reiterate:
\begin_inset Formula $f=0$
\end_inset
corresponds to a pure overlap projection merge: the cluster
\begin_inset Formula $g$
\end_inset
consists of only those disjuncts that are shared in common by
\begin_inset Formula $a$
\end_inset
and
\begin_inset Formula $b$
\end_inset
.
Setting
\begin_inset Formula $f=1$
\end_inset
corresponds to the union merge: all disjuncts on
\begin_inset Formula $a$
\end_inset
and
\begin_inset Formula $b$
\end_inset
are moved into
\begin_inset Formula $g$
\end_inset
so that
\begin_inset Formula $a^{\prime}$
\end_inset
and
\begin_inset Formula $b^{\prime}$
\end_inset
are empty after the merge.
Other fractions
\begin_inset Formula $0<f<1$
\end_inset
interpolate linearly between the overlap merge and the union merge.
The table below shows what happens.
\end_layout
\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="19" columns="7">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
frac
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
total MI
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
MI(g,g)
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
MI(g,of)
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
MI(g,to)
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
p(of,of)
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
p(g,of)
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.0
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
11.312
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
10.499
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $-\infty$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $-\infty$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.312E-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1e-6
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
11.312
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
10.499
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-8.816
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-9.011
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.311E-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.27E-10
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1e-5
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
11.311
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
10.499
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-5.494
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-5.689
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.311E-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.267E-9
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1e-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
11.310
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
10.498
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-2.173
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-2.368
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.310E-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.266E-8
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.001
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.1478
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.9527
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.297E-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.260E-7
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.002
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
11.285
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
10.492
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.1461
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.9510
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.282E-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.450E-6
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.005
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
11.253
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
10.482
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.4630
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.2679
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.239E-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.615E-6
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.01
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
11.207
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
10.465
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_layout
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4.2595
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.166E-4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.194E-6
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_layout
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</cell>
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\begin_layout Plain Layout
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\begin_inset Text
\begin_layout Plain Layout
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\end_layout
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_layout
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</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.452E-5
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_layout
\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
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\begin_layout Plain Layout
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7.312E-8
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7.194E-6
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9.6976
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</cell>
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $-\infty$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $-\infty$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
So, this numeric exploration of the post-merge mutual information reveals
...
something that perhaps should have been obvious, in retrospect.
Sigh.
\end_layout
\begin_layout Standard
In retrospect, it should be clear that the union merge serves only to make
word-sense disambiguation cloudier and mushier, by placing unrelated disjuncts
into the same class/cluster.
This seemed somewhat harmless at some conceptual level, by employing the
argument that the observed disjuncts are rather noisy, and that perhaps
there are commonalities that simply were not observed, due to insufficient
sampling.
This may still be true, but, viewed from the MI angle, it is clear that
anything beyond a pure overlap merge is harmful to the entropy maximization.
\end_layout
\begin_layout Subsection*
Deriving the Entropy Extrema
\end_layout
\begin_layout Standard
The goal of this section is to find the extrema of the relative entropy
as given in eqn
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:relative entropy"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
This requires a lot of algebraic calculation.
The calculations below were undertaken before the numeric exploration immediate
ly above.
The numerics show that this effort was ...
pointless.
The entropy is maximized by doing the overlap merge, setting
\begin_inset Formula $f=0$
\end_inset
; it's it, that's that.
The below is simply not necessary.
It's kept below for, uhh, posterity.
Otherwise, its useless and can be skipped over.
\end_layout
\begin_layout Subsubsection*
Merge parameterization
\end_layout
\begin_layout Standard
It's worth considering the most general merge that maintains detailed balance,
and symmetry between the two merged words.
This is done by defining a merge vector - a merge parameter for each disjunct,
so that one gets a vector
\begin_inset Formula $0\le f\left(d\right)\le1$
\end_inset
.
The merged class is then
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
N\left(g,d\right)=\begin{cases}
f\left(d\right)\left[N\left(a,d\right)+N\left(b,d\right)\right] & d\in D\left(a\right)\cap D\left(b\right)\\
f\left(d\right)N\left(a,d\right) & d\in D\left(a\right)\backslash D\left(b\right)\\
f\left(d\right)N\left(b,d\right) & d\in D\left(b\right)\backslash D\left(a\right)
\end{cases}
\]
\end_inset
The above tries to distinguish the overlapping regions.
But it is overly complicated, since on the non-overlapping regions, the
counts vanish.
That is,
\begin_inset Formula $N\left(b,d\right)=0$
\end_inset
when
\begin_inset Formula $d\in D\left(a\right)\backslash D\left(b\right)$
\end_inset
.
Thus, its enough to write
\begin_inset Formula
\[
N\left(g,d\right)=f\left(d\right)\left[N\left(a,d\right)+N\left(b,d\right)\right]
\]
\end_inset
To maintain detailed balance as before, one must have
\begin_inset Formula
\[
N\left(a^{\prime},d\right)=\left(1-f\left(d\right)\right)N\left(a,d\right)
\]
\end_inset
Note that the number of disjuncts is huge: in practice, a vocabulary of
a few thousands words might have hundreds of thousands of disjuncts.
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
The `run-2` dataset used in the example above had 6029 distinct words and
98102 distinct disjuncts.
This was a
\begin_inset Quotes eld
\end_inset
trimmed
\begin_inset Quotes erd
\end_inset
dataset, 40-8-5: all words with less than 40 observations were discarded.
All disjuncts with less than 8 observations were discarded, and all (word,disju
nct) pairs with less than 5 observations were discarded.
I've been assuming that trimming filters out a lot of the noise in the
dataset, without much compromising the data integrity.
However, this has not been shown experimentally, it just seems like it
should be so.
\end_layout
\end_inset
The vector
\begin_inset Formula $f\left(d\right)$
\end_inset
is a large vector.
\end_layout
\begin_layout Standard
In what follows, this full vector will be retreated from, going back to
the original vision of a projection merge.
There are two reasons for this:
\end_layout
\begin_layout Itemize
The expressions get hopelessly non-linear and intractable,
\end_layout
\begin_layout Itemize
It is hard to imagine what sort of additional information might arrive,
that would distinguish one disjunct from another.
Either a disjunct is associated with
\begin_inset Formula $a$
\end_inset
or with
\begin_inset Formula $b$
\end_inset
or with both; there really aren't any other possibilities.
\end_layout
\begin_layout Standard
However, until that point, a vector of independent
\begin_inset Formula $f$
\end_inset
's will be assumed.
\end_layout
\begin_layout Subsubsection*
Variational problem
\end_layout
\begin_layout Standard
A change of notation is in order.
Parenthesis have been useful to emphasize what depends on what.
Parenthesis are also convenient for plain-ASCII source code documentation.
However, for the following, subscript notation will improve readability.
Let
\begin_inset Formula
\begin{align*}
f_{d} & \equiv f\left(d\right)\\
N_{wd} & \equiv N\left(w,d\right)\\
M_{uw} & \equiv MI\left(u,w\right)
\end{align*}
\end_inset
The variational problem is then to obtain
\begin_inset Formula
\[
\frac{\partial S}{\partial f_{d}}
\]
\end_inset
I see no easy way out, other than to brute-force this.
First, note that
\begin_inset Formula
\begin{align*}
\frac{\partial}{\partial f_{d}}\sum_{u,w=a,b}p\left(u,w\right)MI\left(u,w\right) & =\frac{\partial}{\partial f_{d}}\left[p\left(a,a\right)MI\left(a,a\right)+2p\left(a,b\right)MI\left(a,b\right)+p\left(b,b\right)MI\left(b,b\right)\right]\\
& =0
\end{align*}
\end_inset
since none of these have
\begin_inset Formula $f\left(d\right)$
\end_inset
appearing in the expressions.
Lets blast away.
\begin_inset Formula
\[
\frac{\partial S}{\partial f_{d}}=\sum_{u,w=g,a^{\prime},b^{\prime}}M_{uw}\frac{\partial p_{uw}}{\partial f_{d}}+p_{uw}\frac{\partial M_{uw}}{\partial f_{d}}
\]
\end_inset
So
\begin_inset Formula
\begin{align*}
\frac{\partial p_{gg}}{\partial f_{d}}= & \frac{1}{i_{**}}\frac{\partial i_{gg}}{\partial f_{d}}\\
= & \frac{1}{i_{**}}\frac{\partial}{\partial f_{d}}\sum_{d}N_{gd}N_{gd}\\
= & \frac{2}{i_{**}}f_{d}\left(N_{ad}+N_{bd}\right)^{2}
\end{align*}
\end_inset
and
\begin_inset Formula
\begin{align*}
\frac{\partial p_{ga^{\prime}}}{\partial f_{d}}= & \frac{1}{i_{**}}\frac{\partial}{\partial f_{d}}N_{gd}N_{a^{\prime}d}\\
= & \frac{1}{i_{**}}\left(N_{ad}+N_{bd}\right)N_{ad}\frac{\partial}{\partial f_{d}}f_{d}\left(1-f_{d}\right)\\
= & \frac{1}{i_{**}}\left(N_{ad}+N_{bd}\right)N_{ad}\left(1-2f_{d}\right)
\end{align*}
\end_inset
and
\begin_inset Formula
\begin{align*}
\frac{\partial p_{a^{\prime}a^{\prime}}}{\partial f_{d}}= & \frac{1}{i_{**}}\frac{\partial}{\partial f_{d}}N_{a^{\prime}d}N_{a^{\prime}d}\\
= & \frac{N_{ad}^{2}}{i_{**}}\frac{\partial}{\partial f_{d}}\left(1-f_{d}\right)^{2}\\
= & -2\frac{N_{ad}^{2}}{i_{**}}\left(1-f_{d}\right)
\end{align*}
\end_inset
and
\begin_inset Formula
\begin{align*}
\frac{\partial M_{uw}}{\partial f_{d}}= & \frac{\partial}{\partial f_{d}}\log_{2}\frac{p_{uw}}{p_{u}p_{w}}\\
= & \frac{1}{\log2}\left[\frac{1}{p_{uw}}\frac{\partial p_{uw}}{\partial f_{d}}-\frac{1}{p_{u}}\frac{\partial p_{u}}{\partial f_{d}}-\frac{1}{p_{w}}\frac{\partial p_{w}}{\partial f_{d}}\right]
\end{align*}
\end_inset
so
\begin_inset Formula
\begin{align*}
\frac{\partial p_{g}}{\partial f_{d}}= & \frac{1}{i_{**}}\frac{\partial i_{g*}}{\partial f_{d}}\\
= & \frac{1}{i_{**}}\frac{\partial}{\partial f_{d}}\sum_{d}N_{gd}N_{*d}\\
= & \frac{1}{i_{**}}\frac{\partial}{\partial f_{d}}N_{gd}N_{*d}\\
= & \frac{1}{i_{**}}N_{*d}\left[N_{ad}+N_{bd}\right]
\end{align*}
\end_inset
and
\begin_inset Formula
\begin{align*}
\frac{\partial p_{a^{\prime}}}{\partial f_{d}}= & \frac{1}{i_{**}}\frac{\partial i_{a^{\prime}*}}{\partial f_{d}}\\
= & -\frac{1}{i_{**}}N_{*d}N_{ad}
\end{align*}
\end_inset
\end_layout
\begin_layout Standard
Putting all this together is painfully tedious.
But it must be done.
(Is there some easier way I don't see yet?)
\begin_inset Formula
\begin{align*}
\sum_{u,w=g,a^{\prime},b^{\prime}}p_{uw}\frac{\partial M_{uw}}{\partial f_{d}}= & p_{gg}\frac{\partial M_{gg}}{\partial f_{d}}+p_{a^{\prime}a^{\prime}}\frac{\partial M_{a^{\prime}a^{\prime}}}{\partial f_{d}}+p_{b^{\prime}b^{\prime}}\frac{\partial M_{b^{\prime}b^{\prime}}}{\partial f_{d}}\\
& \quad+2p_{ga^{\prime}}\frac{\partial M_{ga^{\prime}}}{\partial f_{d}}+2p_{gb^{\prime}}\frac{\partial M_{gb^{\prime}}}{\partial f_{d}}\\
= & \frac{1}{\log2}\times\\
& \frac{2}{i_{**}}f_{d}\left(N_{ad}+N_{bd}\right)^{2}-2\frac{p_{gg}}{p_{g}}\cdot\frac{1}{i_{**}}N_{*d}\left(N_{ad}+N_{bd}\right)\\
& -2\frac{N_{ad}^{2}}{i_{**}}\left(1-f_{d}\right)-2\frac{p_{a^{\prime}a^{\prime}}}{p_{a^{\prime}}}\cdot\left(-\frac{1}{i_{**}}N_{*d}N_{ad}\right)\\
& -2\frac{N_{bd}^{2}}{i_{**}}\left(1-f_{d}\right)+2\frac{p_{b^{\prime}b^{\prime}}}{p_{b^{\prime}}}\cdot\left(\frac{1}{i_{**}}N_{*d}N_{bd}\right)\\
& +2\frac{1}{i_{**}}\left(N_{ad}+N_{bd}\right)N_{ad}\left(1-2f_{d}\right)+2\frac{p_{ga^{\prime}}}{p_{a^{\prime}}}\cdot\frac{1}{i_{**}}N_{*d}N_{ad}+2\frac{p_{ga^{\prime}}}{p_{b^{\prime}}}\cdot\frac{1}{i_{**}}N_{*d}N_{bd}\\
& +2\frac{1}{i_{**}}\left(N_{ad}+N_{bd}\right)N_{bd}\left(1-2f_{d}\right)+2\frac{p_{gb^{\prime}}}{p_{a^{\prime}}}\cdot\frac{1}{i_{**}}N_{*d}N_{ad}+2\frac{p_{gb^{\prime}}}{p_{b^{\prime}}}\cdot\frac{1}{i_{**}}N_{*d}N_{bd}
\end{align*}
\end_inset
\end_layout
\begin_layout Standard
Lets hope there are no mistakes.
Gathering terms,
\begin_inset Formula
\begin{align*}
\frac{i_{**}}{2}\log2\sum_{u,w=g,a^{\prime},b^{\prime}}p_{uw}\frac{\partial M_{uw}}{\partial f_{d}}= & f_{d}\left[\left(N_{ad}+N_{bd}\right)^{2}+N_{ad}^{2}+N_{bd}^{2}-2\left(N_{ad}+N_{bd}\right)^{2}\right]\\
& -\frac{p_{gg}}{p_{g}}N_{*d}\left(N_{ad}+N_{bd}\right)+\frac{p_{a^{\prime}a^{\prime}}}{p_{a^{\prime}}}N_{*d}N_{ad}+\frac{p_{b^{\prime}b^{\prime}}}{p_{b^{\prime}}}N_{*d}N_{bd}\\
& -N_{ad}^{2}-N_{bd}^{2}\\
& +\left(N_{ad}+N_{bd}\right)^{2}\\
& +\frac{p_{ga^{\prime}}}{p_{a^{\prime}}}N_{*d}N_{ad}+\frac{p_{ga^{\prime}}}{p_{b^{\prime}}}N_{*d}N_{bd}\\
& +\frac{p_{gb^{\prime}}}{p_{a^{\prime}}}N_{*d}N_{ad}+\frac{p_{gb^{\prime}}}{p_{b^{\prime}}}N_{*d}N_{bd}\\
= & 2\left(1-f_{d}\right)N_{ad}N_{bd}\\
& \quad+N_{*d}N_{ad}\left[-\frac{p_{gg}}{p_{g}}+\frac{p_{a^{\prime}a^{\prime}}+p_{ga^{\prime}}+p_{gb^{\prime}}}{p_{a^{\prime}}}\right]\\
& \quad+N_{*d}N_{bd}\left[-\frac{p_{gg}}{p_{g}}+\frac{p_{b^{\prime}b^{\prime}}+p_{ga^{\prime}}+p_{gb^{\prime}}}{p_{b^{\prime}}}\right]
\end{align*}
\end_inset
Huf.
This appears to be quadratic rational in
\begin_inset Formula $f$
\end_inset
because
\begin_inset Formula
\begin{align*}
p_{gg}= & \frac{1}{i_{**}}\sum_{d}N_{gd}N_{gd}\\
= & \frac{1}{i_{**}}\sum_{d}f_{d}^{2}\left(N_{ad}+N_{bd}\right)^{2}
\end{align*}
\end_inset
while
\begin_inset Formula
\begin{align*}
p_{g}= & \frac{1}{i_{**}}\sum_{d}N_{gd}N_{*d}\\
= & \frac{1}{i_{**}}\sum_{d}f_{d}\left(N_{ad}+N_{bd}\right)N_{*d}
\end{align*}
\end_inset
So those fractions are nasty.
There seem to be two choices in front of us.
These are
\end_layout
\begin_layout Itemize
Set
\begin_inset Formula $f_{d}=1$
\end_inset
for
\begin_inset Formula $d\in D\left(a\right)\cap D\left(b\right)$
\end_inset
and
\begin_inset Formula $f_{d}=f$
\end_inset
otherwise.
This is the projection merge.
\end_layout
\begin_layout Itemize
Set
\begin_inset Formula $f_{d}=f$
\end_inset
for all
\begin_inset Formula $d$
\end_inset
.
This seems to be pointless; it smears together multiple word senses.
\end_layout
\begin_layout Standard
Abandon the second option.
To disentangle the location of
\begin_inset Formula $f$
\end_inset
in the various quantities, some additional notation is needed.
\end_layout
\begin_layout Section*
July 2021 - Bad Merge Redux
\end_layout
\begin_layout Standard
Variation on above theme.
After merging the first two words with the overlap merge, the MI to additional
words gets large, and so merges to those are proposed, too.
This is presumably some bug.
What's the bug? Attempt to find out, below.
\end_layout
\begin_layout Standard
The problem dataset is `
\family typewriter
r2-mpg-trim-40-8-5.rdb
\family default
` (the same trimmed dataset as before, above.) Using the merge strategy `
\family typewriter
(gram-classify-greedy-mifuzz star-obj 5.0 0 4)
\family default
` i.e.
MI cutoff of 5.0, zero fraction and min obs count of 4, the following is
printed:
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="7" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell multicolumn="1" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
(bad) merger MI
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
be
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
have
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
5.5520
\end_layout
\end_inset
</cell>
</row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
be-have
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
him
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
5.6404
\end_layout
\end_inset
</cell>
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
be-have
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
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\end_layout
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\begin_inset Text
\begin_layout Plain Layout
7.8926
\end_layout
\end_inset
</cell>
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\begin_inset Text
\begin_layout Plain Layout
this
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
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\end_layout
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</cell>
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
be-have
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
all
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.0597
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
be-have
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
her
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.4523
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Standard
The be-have merger is plausible.
The remaining ones are not.
\end_layout
\begin_layout Standard
Lets look at the MI before merger.
As before, load `mi-tools.scm` to load scaffolding to make this easy.
It is
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="9" columns="8">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell multicolumn="1" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Symmetric MI - csets+shapes
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
be
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
have
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
him
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
do
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
this
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
all
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
her
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
be
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7.349
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
5.552
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.464
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.107
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-0.29
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.684
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.316
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
have
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
6.922
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.010
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.953
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-0.93
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.242
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-0.24
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
him
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.201
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.830
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.181
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.500
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.961
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
do
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
5.242
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.431
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.723
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.683
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
this
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
6.245
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.223
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.011
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
all
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4.421
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.573
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
her
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.942
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Standard
The above is the MI for disjuncts plus shapes.
It all looks pretty darned reasonable.
\end_layout
\begin_layout Standard
Lets check to see what's going on with the csets-only, without the shapes.
This requires a batch-transpose recompute.
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="9" columns="8">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell multicolumn="1" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Symmetric MI - csets only
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
be
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
have
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
him
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
do
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
this
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
all
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
her
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
be
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
8.709
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
5.643
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4.284
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4.608
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-1.70
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.119
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.909
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
have
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
8.267
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.772
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4.655
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-3.98
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.234
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-0.14
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
him
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Standard
So that's ..
a wash.
The shapes really lowered the MI on him, her.
Made them much much more distant.
It lowered all the self-MI's.
It lowered some cross-MI's and raised others, with no apparent overall
effect.
\end_layout
\begin_layout Subsection*
How Shapes affect MI
\end_layout
\begin_layout Standard
The effect on him-her deserves further pondering.
Without the shapes, it seems to be saying that him-her are being used in
sentences the same way.
But, when shapes are added, and the self-MI drops, this seems to be saying
\begin_inset Quotes eld
\end_inset
hey, besides him/her, there are other words in this cluster, besides just
him/her
\begin_inset Quotes erd
\end_inset
.
That's what I interpret this as saying.
\end_layout
\begin_layout Standard
Lets take a detour, and see if that is true.
Table below shows top MI between
\begin_inset Quotes eld
\end_inset
him
\begin_inset Quotes erd
\end_inset
and other words, ranked by cset-only-MI.
A second column shows the corresponding cset-only-MI to
\begin_inset Quotes eld
\end_inset
her
\begin_inset Quotes erd
\end_inset
, although this is no longer ranked.
The third column shows the cset+shape-MI to
\begin_inset Quotes eld
\end_inset
him
\begin_inset Quotes erd
\end_inset
.
\end_layout
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\align center
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\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
It's OK, but a bit underwhelming.
First, looking at the first column.
It seem OK that other pronouns are in there.
The presence of a handful of verbs, nouns and prepositions is not confidence-in
spiring.
One might have hoped that these would not be there.
The second column is OK, I guess...
although
\begin_inset Quotes eld
\end_inset
her
\begin_inset Quotes erd
\end_inset
is also a possessive, and
\begin_inset Quotes eld
\end_inset
him
\begin_inset Quotes erd
\end_inset
is not.
\end_layout
\begin_layout Standard
The third column is peculiar.
It looks at the cset+shape MI.
It's a
\emph on
lot
\emph default
lower.
I'm surprised by this; I was expecting the MI to be improved.
There's only a hint of a silver lining: the MI to verbs, nouns and preps
seems just a little lower, than it is to pronouns.
But just a smidgen.
It's not much.
Overall, we conclude that the MI values fail to distinguish between pronouns,
verbs, nouns and preps.
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
So this is intermixed with a lot of confusion.
The MM^T marginals (
\emph on
i.e.
\emph default
the values of
\begin_inset Formula $\sum_{d}N(w,d)N(*,d)$
\end_inset
) were originally from the large dataset, kept as a cache.
Then the dataset was trimmed, without recomputing marginals.
Then I caught the error.
Then I discovered that the
\begin_inset Formula $N(*,d)$
\end_inset
values were cached, too.
So, several different mistakes – in some cases, the sum ran over the set
of disjuncts
\begin_inset Formula $d$
\end_inset
for the full set, and for some it did not.
The dot-product
\begin_inset Formula $\sum_{d}N(w,d)N(u,d)$
\end_inset
used in computing the MI always runs only over the disjuncts
\begin_inset Formula $d$
\end_inset
in the trimmed dataset.
This is obviously not compatible with the cached marginals.
\end_layout
\end_inset
\end_layout
\begin_layout Section*
July 2021 - Vector Sizes and MI
\end_layout
\begin_layout Standard
Above, we see that, for a given dataset, the MI values (seem to) fail to
distinguish between pronouns, verbs, nouns and prepositions.
This section explores why that is the case.
The results so far show that the MI between a large word-vector and a small
word-vector can be quite high, if most of the small (
\begin_inset Quotes eld
\end_inset
short
\begin_inset Quotes erd
\end_inset
) vector lies in the same direction as the large (
\begin_inset Quotes eld
\end_inset
long
\begin_inset Quotes erd
\end_inset
) vector.
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
The word
\begin_inset Quotes eld
\end_inset
length
\begin_inset Quotes erd
\end_inset
is troublesome.
In this paragraph, we are talking about
\begin_inset Quotes eld
\end_inset
the size of the support
\begin_inset Quotes erd
\end_inset
\emph on
i.e.
\emph default
the number of non-zero entries in the vector.
This should not be confused with the root-mean-square length of the vector,
\emph on
i.e.
\emph default
its Euclidean length.
This confusion easily extends to
\begin_inset Quotes eld
\end_inset
short
\begin_inset Quotes erd
\end_inset
vs.
\begin_inset Quotes eld
\end_inset
small
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
long
\begin_inset Quotes erd
\end_inset
vs.
\begin_inset Quotes eld
\end_inset
large
\begin_inset Quotes erd
\end_inset
.
Hopefully, things will be clear from context.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
This is...well, its good and bad, depending on what one wants.
If the goal is to take a short fragment and identify where it belongs,
then this is unalloyed good.
However, if the dataset is small, and almost all words have short vectors,
then this is bad.
The natural Zipfian distribution indicates that only a small fraction (ballpark
of 10%) have long vectors.
Thus, it is not surprising that most short vectors will mostly align with
one of these long vectors.
\end_layout
\begin_layout Standard
There are several conclusions one can make:
\end_layout
\begin_layout Itemize
Don't use small datasets with lots of short vectors.
This is hard, because (1) the Zipfian distribution means most vectors will
be short, no matter the size of the dataset.
(2) it is CPU intensive to use large datasets.
\end_layout
\begin_layout Itemize
Don't run merge algos on short vectors.
This is already done, more or less: the current merge algo considers merging
only vectors of similar counts (and thus, presumably of similar sizes.)
Stop merging long before reaching down into the short-vector regions of
the dataset.
\end_layout
\begin_layout Itemize
Don't trim datasets.
The goal of trimming was to discard
\begin_inset Quotes eld
\end_inset
noise
\begin_inset Quotes erd
\end_inset
: i.e.
infrequently-observed word-disjunct pairs, with the general argument that
such low-count pairs are likely to be garbage.
However, we have not shown that they actually are
\begin_inset Quotes eld
\end_inset
garbage
\begin_inset Quotes erd
\end_inset
, and such low-count trimming might actually be discarding important information.
That is, trimming might be creating short vectors that incorrectly align
with large vectors, when the untrimmed variants would have shown that they
are quite different.
\end_layout
\begin_layout Itemize
Use a different measure of MI that emphasizes non-colinearity.
\end_layout
\begin_layout Subsection*
Terrible MI values
\end_layout
\begin_layout Standard
Here's the disaster were looking at.
The top-MI cset+shape words to
\begin_inset Quotes eld
\end_inset
him
\begin_inset Quotes erd
\end_inset
are these:
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
Use `(almi
\begin_inset Quotes eld
\end_inset
him
\begin_inset Quotes erd
\end_inset
)` from the tools file.
\end_layout
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
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...14 more ...
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\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
Wow.
That's ugly.
Bad.
Wrong.
None of the skipped 14 words are pronouns.
The self-MI for
\begin_inset Quotes eld
\end_inset
him
\begin_inset Quotes erd
\end_inset
is 25th on the list.
Why isn't the self-MI always the highest? How should this be understood?
\end_layout
\begin_layout Standard
The word
\begin_inset Quotes eld
\end_inset
her
\begin_inset Quotes erd
\end_inset
is no better:
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
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<features tabularvalignment="middle">
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\begin_inset Text
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...
7 more ...
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\begin_inset Text
\begin_layout Plain Layout
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<row>
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\begin_inset Text
\begin_layout Plain Layout
her
\end_layout
\end_inset
</cell>
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\begin_inset Text
\begin_layout Plain Layout
2.878
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</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
Here, the self-MI for
\begin_inset Quotes eld
\end_inset
her
\begin_inset Quotes erd
\end_inset
is 17th in the list.
At least the possessive
\begin_inset Quotes eld
\end_inset
his
\begin_inset Quotes erd
\end_inset
appears 8th.
\end_layout
\begin_layout Subsection*
Close-up inspection
\end_layout
\begin_layout Standard
So what is going on here? Lets look close up.
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
After the analysis in this section was performed, a procedural issue was
detected.
It's not clear how this affects results.
The procedural problem is that the MM^T marginals were computed first,
and then the dataset was trimmed.
For computing the MI, the MM^T marginals from the full dataset were used.
I'm thinking that they should have been recomputed after trimming.
\end_layout
\end_inset
The word
\begin_inset Quotes eld
\end_inset
him
\begin_inset Quotes erd
\end_inset
has 404 Sections and 430 CrossSections.
A random sample of the CrossSections looks mighty healthy:
\end_layout
\begin_layout Standard
\begin_inset listings
lstparams "basicstyle={\ttfamily}"
inline false
status open
\begin_layout Plain Layout
at: (gazing- & him+) or (gazing- & the+)
\end_layout
\begin_layout Plain Layout
upon: (came- & him+) or (came- & a+)
\end_layout
\begin_layout Plain Layout
for: impossible- & (him+ or me+ or any+ or them+)
\end_layout
\begin_layout Plain Layout
do: to- & (him+ or her+ or it anything in a something them what ? so
with as the that this .
, )
\end_layout
\end_inset
\end_layout
\begin_layout Standard
The Sections look good too:
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
him: (assure- & that+) or (put- & in+) or (Let- & go+) or (turned- &out+)
\end_layout
\end_inset
\end_layout
\begin_layout Standard
How about the word
\begin_inset Quotes eld
\end_inset
sink
\begin_inset Quotes erd
\end_inset
? Wow, it is ..
bad.
It has one cross-section, and zero sections:
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
It has zero sections because these were filtered out.
The filter removes all sections with a count of less than five.
It seems that the word
\begin_inset Quotes eld
\end_inset
sink
\begin_inset Quotes erd
\end_inset
only appeared in a small handful of sentences.
This makes it a rare word.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
to: seemed- & sink+
\end_layout
\end_inset
\end_layout
\begin_layout Standard
The total observation count is 9, which is small.
Just above the dataset cutoff.
The self-MI is MI(sink, sink)=12.27.
The cross section is
\begin_inset Quotes eld
\end_inset
seemed to xxx
\begin_inset Quotes erd
\end_inset
where xxx can be any of
\begin_inset Quotes eld
\end_inset
like them Pierre grow her have go consider see be take feel say think him
make come sink fill me
\begin_inset Quotes erd
\end_inset
.
Count on
\begin_inset Quotes eld
\end_inset
seemed to him
\begin_inset Quotes erd
\end_inset
is 151.
Count on the product is
\begin_inset Formula $\sum_{d}N(sink,d)N(him,d)=9\times151=1359$
\end_inset
.
So that's perfect.
\begin_inset Formula $\sum_{d}N(sink,d)N(*,d)=10215$
\end_inset
Since there is one d, we have
\begin_inset Formula $\sum_{d}N(*,d)=1135$
\end_inset
which was verified manually.
The other marginal is
\begin_inset Formula $\sum_{d}N(him,d)N(*,d)=41576753$
\end_inset
which was not verified.
The total count is 6842054908.
Multiplying out:
\begin_inset Formula
\begin{align*}
\log_{2}\frac{\sum_{d}N(sink,d)N(him,d)}{\sum_{d}N(*,d)N(*,d)} & =\log_{2}\frac{1359}{6842054908}=-22.263\\
\log_{2}\frac{\sum_{d}N(sink,d)N(*,d)}{\sum_{d}N(*,d)N(*,d)} & =\log_{2}\frac{10215}{6842054908}=-19.353\\
\log_{2}\frac{\sum_{d}N(him,d)N(*,d)}{\sum_{d}N(*,d)N(*,d)} & =\log_{2}\frac{41576753}{6842054908}=-7.362
\end{align*}
\end_inset
and so
\begin_inset Formula
\[
MI(sink,him)=-22.263+19.353+7.362=4.452
\]
\end_inset
So the numerics verify.
What does this teach us? A word with a single cross-section can give us
absurd MI values.
\end_layout
\begin_layout Standard
Lets compare this to the cosine distance.
For that, we need:
\begin_inset Formula
\begin{align*}
\sum_{d}N(sink,d)N(him,d) & =1359\\
\sum_{d}N(sink,d)N(sink,d) & =81\\
\sum_{d}N(him,d)N(him,d) & =1161813
\end{align*}
\end_inset
and the cosine would be
\begin_inset Formula
\begin{align*}
\cos\theta & =\frac{\sum_{d}N(sink,d)N(him,d)}{\sqrt{\left(\sum_{d}N(sink,d)N(sink,d)\right)\left(\sum_{d}N(him,d)N(him,d)\right)}}\\
& =\frac{1359}{\sqrt{81\times1161813}}\\
& =0.1401
\end{align*}
\end_inset
which is a far more reasonable value.
Why is the MI misleading us? What went wrong here? What can we do to fix
it?
\end_layout
\begin_layout Subsection*
What's the problem?
\end_layout
\begin_layout Standard
Well, the primary problem with the above is that ..
the MI is correct, but unwanted.
We have a word with a tiny sample - one disjunct, and that one disjunct
overlaps nicely with anything and everything else that has that disjunct
in it.
\end_layout
\begin_layout Standard
To with: the cross-section came from the phrase
\begin_inset Quotes eld
\end_inset
seemed to xxx
\begin_inset Quotes erd
\end_inset
.
It's plausible to assume that
\begin_inset Formula $MI(sink,xxx)$
\end_inset
will be high for all other words xxx appearing in that phrase.
Indeed this is the case:
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Box Frameless
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hor_pos "c"
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status open
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\begin_inset Text
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\end_inset
\end_layout
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
The column labeled
\begin_inset Quotes eld
\end_inset
support
\begin_inset Quotes erd
\end_inset
is the length of the vector for that word (
\emph on
i.e.
\emph default
it is the number of disjuncts observed for that word.) It appears to show
that the length of the vector has little impact on the MI.
That is, it is the word
\begin_inset Quotes eld
\end_inset
sink
\begin_inset Quotes erd
\end_inset
and it's single disjunct that is in the drivers seat: it can overlap some
very long vectors in such a way that a high MI results.
\end_layout
\begin_layout Standard
What is meant by
\begin_inset Quotes eld
\end_inset
high MI
\begin_inset Quotes erd
\end_inset
? (Computing the average MI is impossible, as the CPU demand to sum over
all pairs is simply too high.
We need to develop techniques for estimating this from the distribution.)
\end_layout
\begin_layout Standard
The MI is high, because this word
\begin_inset Quotes eld
\end_inset
sink
\begin_inset Quotes erd
\end_inset
has nothing else it can match to.
That is, the vector for
\begin_inset Quotes eld
\end_inset
sink
\begin_inset Quotes erd
\end_inset
is so short, that it matches anything else that has that same vector element
in it.
So, in that sense, the result is correct; it is just, perhaps, not quite
what we wanted.
\end_layout
\begin_layout Standard
What do we want, then?
\end_layout
\begin_layout Subsection*
Double check
\end_layout
\begin_layout Standard
The above focuses on the word
\begin_inset Quotes eld
\end_inset
sink
\begin_inset Quotes erd
\end_inset
.
Perhaps its exceptional.
How about the other problematic words:
\begin_inset Quotes eld
\end_inset
fill Instead grow 1880 scoundrel accepted serve appealed springing
\begin_inset Quotes erd
\end_inset
? Let's take a quick peak.
\end_layout
\begin_layout Standard
The word
\begin_inset Quotes eld
\end_inset
fill
\begin_inset Quotes erd
\end_inset
shows up in two cross-sections:
\begin_inset Quotes eld
\end_inset
to: seemed- & fill+
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
up: fill- & the+
\begin_inset Quotes erd
\end_inset
, both with low counts (9 and 10) and no Sections.
\end_layout
\begin_layout Standard
The word
\begin_inset Quotes eld
\end_inset
Instead
\begin_inset Quotes erd
\end_inset
appears in one Section:
\begin_inset Quotes eld
\end_inset
Instead: WALL- & ,+ & he+
\begin_inset Quotes erd
\end_inset
and one cross-section:
\begin_inset Quotes eld
\end_inset
,: Instead- & he+
\begin_inset Quotes erd
\end_inset
with counts of 6 and 14.
\end_layout
\begin_layout Standard
The word
\begin_inset Quotes eld
\end_inset
grow
\begin_inset Quotes erd
\end_inset
appears in three cross sections:
\begin_inset Quotes eld
\end_inset
to: seemed- & grow+
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
up: grow- & .+
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
to: began- & grow+
\begin_inset Quotes erd
\end_inset
with counts of 18, 6 and 25.
There are no Sections.
\end_layout
\begin_layout Standard
The word
\begin_inset Quotes eld
\end_inset
1880
\begin_inset Quotes erd
\end_inset
appears in one Section:
\begin_inset Quotes eld
\end_inset
1880: in- & ,+
\begin_inset Quotes erd
\end_inset
with a count of 10.
\end_layout
\begin_layout Standard
The word
\begin_inset Quotes eld
\end_inset
scoundrel
\begin_inset Quotes erd
\end_inset
appears in two cross-sections:
\begin_inset Quotes eld
\end_inset
a: am- & scoundrel+
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
,: scoundrel- & but+
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Standard
The word
\begin_inset Quotes eld
\end_inset
accepted
\begin_inset Quotes erd
\end_inset
appears in a number of Sections and cross-sections.
Most of these appear to be the result of parsing a sentence in the Project
Gutenberg boilerplate, something about accepting donations.
\end_layout
\begin_layout Standard
The word
\begin_inset Quotes eld
\end_inset
serve
\begin_inset Quotes erd
\end_inset
appears in one cross-section:
\begin_inset Quotes eld
\end_inset
as: serve- & a+
\begin_inset Quotes erd
\end_inset
with a count of 32.
\end_layout
\begin_layout Standard
We can safely conclude that
\begin_inset Quotes eld
\end_inset
grow
\begin_inset Quotes erd
\end_inset
is prototypical of the other problematic words.
All of the problematic words are those with very short vectors.
\end_layout
\begin_layout Standard
As noted before, the lack of Sections is due to this being a filtered dataset:
Sections with a count of 5 or less have been removed.
Effectively, all of the above words are rare, and occur only in a handful
of sentences in the text.
\end_layout
\begin_layout Standard
This reinforces an apparent paradox: the corpus seems to be quite large:
maybe a quarter of a million sentences! (The exact count is not known,
this is a damaged dataset.) Even after filtering, there seem to be a lot
of words (just over six thousand!) Yet, there's an average of 23 observations
per word, which is not all that many.
This should be compared to mother-baby talk in human infants: by the age
of two, a baby will have heard a small vocabulary of hundreds of words,
but with a sample of a hundred-thousand or a million for each of these
words.
\end_layout
\begin_layout Standard
The size of the datasets used here is deceiving.
Also: one can presume that almost all sentences in the Project Gutenberg
corpus are distinct, with very few repeated sentences.
By contrast, in human mother-baby talk, there is a vast amount of repetition.
Now, most of this repetition is for auditory training, not syntactic training,
but still, the repetitiveness must surely be quite reinforcing.
\end_layout
\begin_layout Subsection*
Other kinds of mutual information
\end_layout
\begin_layout Standard
The above suggests that perhaps other measures should be examined.
Intuitively, one might want a measure that treats vectors of similar size
as being more similar, than those of dis-similar sizes.
\end_layout
\begin_layout Standard
(Conclusion for this section: (1) the conventional MI seems to work best,
and (2) never got around to making a size-sensitive distance in this particular
section.)
\end_layout
\begin_layout Standard
Since we're discarding cosine distance for
\emph on
a priori
\emph default
reasons (probabilities do not live in Euclidean spaces), then only other
MI-like constructions are allowed.
One obvious one is a Banach-like
\begin_inset Formula $MI_{s}$
\end_inset
measure:
\begin_inset Formula
\begin{align*}
MI_{s,t}\left(w,u\right) & =\log_{2}\frac{\left(\sum_{d}N^{s}(w,d)N^{s}(u,d)\right)\left(\sum_{d}N^{t}(*,d)N^{t}(*,d)\right)}{\left(\sum_{d}N^{s}(w,d)N^{t}(*,d)\right)\left(\sum_{d}N^{s}(u,d)N^{t}(*,d)\right)}\\
& =\log_{2}\frac{\left(\sum_{d}p^{s}(w,d)p^{s}(u,d)\right)\left(\sum_{d}p^{t}(*,d)p^{t}(*,d)\right)}{\left(\sum_{d}p^{s}(w,d)p^{t}(*,d)\right)\left(\sum_{d}p^{s}(u,d)p^{t}(*,d)\right)}
\end{align*}
\end_inset
where
\begin_inset Formula $p\left(w,d\right)\equiv N\left(w,d\right)/N\left(*,*\right)$
\end_inset
or where
\begin_inset Formula $p\left(w,d\right)\equiv N\left(w,d\right)/N\left(w,*\right)$
\end_inset
as desired; the normalization cancels out.
The two exponents
\begin_inset Formula $s$
\end_inset
and
\begin_inset Formula $t$
\end_inset
are independent of one-another.
it's not entirely clear if they should be forced to be equal,
\emph on
i.e.
\emph default
to set
\begin_inset Formula $s=t$
\end_inset
or if it is better to set
\begin_inset Formula $t=1$
\end_inset
and allowing
\begin_inset Formula $s$
\end_inset
to remain free.
Even setting
\begin_inset Formula $t=0$
\end_inset
is plausible.
Setting
\begin_inset Formula $s=t=1$
\end_inset
gives the earlier MI.
Setting
\begin_inset Formula $s=t=0$
\end_inset
gives a Hamming-distance-like MI that only considers those disjuncts that
have a non-zero count.
Setting
\begin_inset Formula $s=t=2$
\end_inset
results in a weighting where the high-count disjuncts dominate, and thus
naturally suppresses
\begin_inset Quotes eld
\end_inset
noise
\begin_inset Quotes erd
\end_inset
: the word-disjunct pairs with low observation counts.
This assumes that low-count word-disjunct pairs actually are
\begin_inset Quotes eld
\end_inset
noise
\begin_inset Quotes erd
\end_inset
.
This has
\emph on
not
\emph default
been experimentally established.
In fact, one can go in the opposite direction: perhaps these carry some
kind of important signal.
In this case, a weighting of
\begin_inset Formula $s=t=1/2$
\end_inset
may provide a good intermediate balance between
\begin_inset Formula $s=t=0$
\end_inset
and
\begin_inset Formula $s=t=1$
\end_inset
.
There's a strange intellectual appeal to an exponent of 1/2, in that it
is
\begin_inset Quotes eld
\end_inset
quantum mechanical
\begin_inset Quotes erd
\end_inset
: it is the square root of the probability.
That this is meaningful in a statistical realm is demonstrated by the Fisher
information metric, which reduces to the flat-space Euclidean metric when
working with the square-root.
This is
\begin_inset Quotes eld
\end_inset
interesting
\begin_inset Quotes erd
\end_inset
, the meaning of which is not yet entirely clear to this author.
(Asking about complex values of
\begin_inset Formula $s$
\end_inset
seems like a step too far, at this time.)
\end_layout
\begin_layout Standard
How might this work out for
\begin_inset Quotes eld
\end_inset
sink
\begin_inset Quotes erd
\end_inset
?
\end_layout
\begin_layout Standard
\align center
\begin_inset VSpace defskip
\end_inset
\begin_inset Tabular
<lyxtabular version="3" rows="13" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $D\left(w,*\right)$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $MI_{0}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $MI_{0.5}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $MI_{1}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $MI_{2}$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
him
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
834
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
5.657
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
28.954
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.201
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-13.495
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
me
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
612
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4.382
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
27.300
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.860
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-13.661
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
example
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
12
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-0.056
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
19.292
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.179
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-8.622
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
us
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
153
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.250
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
25.031
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.759
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-11.774
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
them
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
372
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4.235
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
26.589
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.884
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-12.781
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
speak
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
58
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.271
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
21.562
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.061
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-10.114
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
instance
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
19
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.295
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
20.724
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.114
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-13.463
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
happen
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
17
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.605
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
19.379
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.800
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-9.381
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
difference
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
20
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-0.174
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
18.974
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.653
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-8.862
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
themselves
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
17
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.683
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
21.598
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.785
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-9.302
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
himself
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
95
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.671
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
24.065
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.817
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-10.951
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
say
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
257
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.709
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
23.289
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.136
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-11.831
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
The above table shows four different kinds of MI between
\begin_inset Quotes eld
\end_inset
him
\begin_inset Quotes erd
\end_inset
and the other words shown.
The
\begin_inset Formula $D\left(w,*\right)$
\end_inset
is the support for that word vector.
Recall the definition
\begin_inset Formula $D\left(w,d\right)=1$
\end_inset
if
\begin_inset Formula $N\left(w,d\right)>0$
\end_inset
and so
\begin_inset Formula $D\left(w,*\right)$
\end_inset
is just the sum – the size of the vector.
\end_layout
\begin_layout Standard
All of these seem to discriminate, but weakly.
That is, its plausible to have a high MI between
\begin_inset Quotes eld
\end_inset
him me us them themselves himself
\begin_inset Quotes erd
\end_inset
and a lower MI to
\begin_inset Quotes eld
\end_inset
example speak instance happen difference say
\begin_inset Quotes erd
\end_inset
and all of these do this.
The discrimination is imperfect.
For
\begin_inset Formula $MI_{0}$
\end_inset
\begin_inset Quotes eld
\end_inset
themselves
\begin_inset Quotes erd
\end_inset
is lower than
\begin_inset Quotes eld
\end_inset
instance speak say
\begin_inset Quotes erd
\end_inset
, so these two clusters overlap.
That's not good.
For
\begin_inset Formula $MI_{1}$
\end_inset
the lowest pronoun is
\begin_inset Quotes eld
\end_inset
us
\begin_inset Quotes erd
\end_inset
at 1.76 while the highest other word is
\begin_inset Quotes eld
\end_inset
example
\begin_inset Quotes erd
\end_inset
at 1.18, so these two clusters are separated.
However, the information content is not large: MI is measured in bits,
there is less than one bit difference.
This is not big enough to get the clustering to work well, as currently
designed/implemented.
\end_layout
\begin_layout Standard
The huge values for
\begin_inset Formula $MI_{0.5}$
\end_inset
are surprising.
It doesn't seem to be a great discriminator: the width of the clusters
is about the same as the distance between them.
The lowest pronoun for
\begin_inset Formula $MI_{0.5}$
\end_inset
is
\begin_inset Quotes eld
\end_inset
themselves
\begin_inset Quotes erd
\end_inset
at 21.60; the highest non-pronoun is
\begin_inset Quotes eld
\end_inset
speak
\begin_inset Quotes erd
\end_inset
at 21.56.
So this seems weaker than the usual MI for discrimination.
\end_layout
\begin_layout Standard
The negative values for
\begin_inset Formula $M_{2}$
\end_inset
are surprising.
Not sure what to make of that.
\end_layout
\begin_layout Standard
Table below shows cluster statistics.
The pronoun cluster is
\begin_inset Quotes eld
\end_inset
him me us them themselves himself
\begin_inset Quotes erd
\end_inset
the
\begin_inset Quotes eld
\end_inset
other
\begin_inset Quotes erd
\end_inset
cluster is
\begin_inset Quotes eld
\end_inset
example speak instance happen difference say
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="17" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
pronouns
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
other
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
min
\begin_inset Formula $MI_{0}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.6833
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-0.174
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
max
\begin_inset Formula $MI_{0}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
5.6577
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.2954
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
avg
\begin_inset Formula $MI_{0}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.4799
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.9418
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
rms
\begin_inset Formula $MI_{0}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.5628
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.9021
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
min
\begin_inset Formula $MI_{0.5}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
21.598
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
18.974
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
max
\begin_inset Formula $MI_{0.5}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
28.955
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
23.289
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
avg
\begin_inset Formula $MI_{0.5}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
25.590
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
20.537
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
rms
\begin_inset Formula $MI_{0.5}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.3741
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.5272
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
min
\begin_inset Formula $MI_{1}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.7585
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.1135
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
max
\begin_inset Formula $MI_{1}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.2012
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.1789
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
avg
\begin_inset Formula $MI_{1}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.8844
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.8238
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
rms
\begin_inset Formula $MI_{1}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.1479
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.3683
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
min
\begin_inset Formula $MI_{2}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-13.66
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-13.46
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
max
\begin_inset Formula $MI_{2}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-9.302
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-8.622
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
avg
\begin_inset Formula $MI_{2}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-11.99
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
-10.38
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
rms
\begin_inset Formula $MI_{2}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.5296
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.7358
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
These are small, eyeballed clusters, but they provide a guide to discrimination.
Consider the difference between the averages in the two clusters, and the
sum of the rms of the two, and the ratio of these:
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="4">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
distance
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
sum rms
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
stddev
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $MI_{0}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.5381
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.4649
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.03
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $MI_{0.5}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
5.053
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.9013
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.30
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $MI_{1}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.0606
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.5162
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.05
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $MI_{2}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.610
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.2654
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.49
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
Thus we conclude – for this small, unbalanced,
\emph on
ad hoc
\emph default
sample, the conventional MI provides the greatest amount of discrimination.
\end_layout
\begin_layout Subsection*
Length Matching
\end_layout
\begin_layout Standard
Above section has no considerations for negative matching – mismatches due
to size differences.
That is, the foundation-stone is built on
\begin_inset Formula
\[
\sum_{d}N\left(w,d\right)N\left(u,d\right)
\]
\end_inset
where, by definition, the sum over the disjuncts
\begin_inset Formula $d$
\end_inset
run over the set where both counts
\begin_inset Formula $N$
\end_inset
are non-zero.
This suggests that perhaps, we could use a measure that includes negative
matching.
This suggests using an MI inspired by the Jaccard distance (aka Tanimoto
distance).
\end_layout
\begin_layout Standard
Is this really needed? Maybe not – and here's why.
Merging words into clusters (minimally) means that only the disjuncts in
the intersection are merged.
The intent of this minimalist merge is to split up words having multiple
meanings, associating a collection of disjuncts with a particular meaning.
But what this also implies is that having a similarity measure that is
likewise tuned to being able to pick out common subsets is a good thing.
\end_layout
\begin_layout Standard
That said, it is worth exploring some alternate similarity measures in greater
detail.
\end_layout
\begin_layout Subsubsection*
Flavors of Jaccard similarity
\end_layout
\begin_layout Standard
There are multiple variants of Jaccard similarity that apply to the current
situation.
This leads to a bit of a naming crisis.
\end_layout
\begin_layout Subsubsection*
Ruzicka similarity
\end_layout
\begin_layout Standard
Given some arbitrary function
\begin_inset Formula $f\left(w,d\right)$
\end_inset
of word-disjunct pairs
\begin_inset Formula $\left(w,d\right)$
\end_inset
, the Ruzicka similarity for this problem is
\begin_inset Formula
\[
J_{R}\left(f;u,w\right)=\frac{\sum_{d}\min\left(f\left(w,d\right),f\left(u,d\right)\right)}{\sum_{d}\max\left(f\left(w,d\right),f\left(u,d\right)\right)}
\]
\end_inset
The Jaccard similarity corresponds to the case where
\begin_inset Formula $f\left(w,d\right)$
\end_inset
is either zero or one (is the indicator function: is or is not in the set)
so the Ruzicka similarity is a kind of weighted Jaccard.
\end_layout
\begin_layout Standard
The corresponding distance is
\begin_inset Formula
\[
J\left(f;u,w\right)=1-J_{R}\left(f;u,w\right)
\]
\end_inset
Wikipedia suggests that this obeys the triangle inequality, i.e.
is a true metric, but this is not obvious to me.
\end_layout
\begin_layout Standard
There are several plausible functions from which
\begin_inset Formula $f$
\end_inset
can be built.
One is the raw observation count
\begin_inset Formula $N\left(w,d\right)$
\end_inset
and the other is the conditional probability of observing a particular
word-disjunct pair, for that word:
\begin_inset Formula
\[
p\left(d|w\right)=\frac{N\left(w,d\right)}{N\left(w,*\right)}
\]
\end_inset
This is nicer than just using the plain counts, and here's why.
Consider first:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
J_{R}\left(N;u,w\right)=\frac{\sum_{d}\min\left(N\left(w,d\right),N\left(u,d\right)\right)}{\sum_{d}\max\left(N\left(w,d\right),N\left(u,d\right)\right)}
\]
\end_inset
If word
\begin_inset Formula $u$
\end_inset
is a (perfect) synonym for
\begin_inset Formula $w$
\end_inset
but is used only rarely, then the comparison of the raw counts will lead
to mostly-garbage.
That is, if
\begin_inset Formula $N\left(u,d\right)=\alpha N\left(w,d\right)$
\end_inset
for some constant
\begin_inset Formula $\alpha$
\end_inset
, these two vectors would come out as dis-similar, even though it is clear
that, for this case, we want them to be considered to be identical, as
they are synonyms.
Thus, using raw
\begin_inset Formula $N$
\end_inset
is just wrong, for the present case.
By contrast, using the conditional probability appears to provide the desired
result: the two synonyms would be judged to have perfect similarity.
\end_layout
\begin_layout Standard
The simplest, narrowest, original definition of the Jaccard similarity is
the unweighted similarity given by
\begin_inset Formula $f\left(w,d\right)=\left[N\left(w,d\right)>0\right]$
\end_inset
.
This just uses the support of
\begin_inset Formula $N$
\end_inset
to determine set membership.
\end_layout
\begin_layout Standard
The `add-similarity-compute` object provides three Jaccard methods:
\end_layout
\begin_layout Itemize
`jaccard` which implements
\begin_inset Formula $J_{R}\left(N;u,w\right)$
\end_inset
\end_layout
\begin_layout Itemize
`cond-jacc` which implements
\begin_inset Formula $J_{R}\left(p;u,w\right)$
\end_inset
\end_layout
\begin_layout Itemize
`overlap` which implements
\begin_inset Formula $J_{R}\left(N>0;u,w\right)$
\end_inset
\end_layout
\begin_layout Subsubsection*
Probability Jaccard
\end_layout
\begin_layout Standard
The
\begin_inset Quotes eld
\end_inset
probability Jaccard
\begin_inset Quotes erd
\end_inset
provides
\begin_inset Quotes eld
\end_inset
maximally consistent sampling
\begin_inset Quotes erd
\end_inset
per WP ref.
It is defined as
\begin_inset Formula
\[
J_{P}\left(u,w\right)=\sum_{d}\left[\sum_{c}\max\left(\frac{N\left(w,c\right)}{N\left(w,d\right)},\frac{N\left(u,c\right)}{N\left(u,d\right)}\right)\right]^{-1}
\]
\end_inset
Note that
\begin_inset Formula
\[
1-J_{P}\left(u,w\right)
\]
\end_inset
is a
\emph on
bona fide
\emph default
metric over the probability space.
\end_layout
\begin_layout Standard
How might this work for
\begin_inset Quotes eld
\end_inset
sink
\begin_inset Quotes erd
\end_inset
with one disjunct, and
\begin_inset Quotes eld
\end_inset
him
\begin_inset Quotes erd
\end_inset
, with 832 disjuncts on it? The sum over
\begin_inset Formula $d$
\end_inset
must necessarily run over just this one disjunct.
The sum over
\begin_inset Formula $c$
\end_inset
runs over all 832 of them.
For 831 of them,
\begin_inset Formula $N(sink,c)=0$
\end_inset
.
Thus,
\begin_inset Formula
\[
J_{P}\left(sink,him\right)=\left[1+\sum_{c\ne d}\frac{N\left(him,c\right)}{N\left(him,d\right)}\right]^{-1}
\]
\end_inset
and clearly, this will judge these two words to be very different.
\end_layout
\begin_layout Subsubsection*
Entropic Similarity
\end_layout
\begin_layout Standard
The notation for entropy and MI being used here are susceptible to confusion,
when being compared to the contents of Wikipedia.
To avoid confusion, a review is worthwhile.
This was already presented in part one of the diary, section titled
\begin_inset Quotes eld
\end_inset
Entropic Similarity
\begin_inset Quotes erd
\end_inset
, but is repeated here.
\end_layout
\begin_layout Standard
Given a count
\begin_inset Formula $N\left(w,d\right)$
\end_inset
define the
\emph on
joint probability
\emph default
\begin_inset Formula
\[
p_{M}\left(w,d\right)=\frac{N\left(w,d\right)}{N\left(*,*\right)}
\]
\end_inset
The subscript
\begin_inset Formula $M$
\end_inset
is a reminder that these counts are obtained from MST/MPG parsing (maximum
spanning tree/maximum planar graph parsing).
\end_layout
\begin_layout Standard
The
\emph on
conditional probability
\emph default
of observing a word
\begin_inset Formula $w$
\end_inset
given a fixed disjunct
\begin_inset Formula $d$
\end_inset
is then
\begin_inset Formula
\[
p_{M}\left(w|d\right)=\frac{p_{M}\left(w,d\right)}{p_{M}\left(*,d\right)}
\]
\end_inset
That is, the conditional probability is just the fraction of all pairs having
this disjunct.
\end_layout
\begin_layout Standard
The
\emph on
marginal probability
\emph default
of observing a word
\begin_inset Formula $w$
\end_inset
is just the sum of observing it for any disjunct:
\begin_inset Formula
\[
p_{M}\left(w\right)\equiv p_{M}\left(w,*\right)=\sum_{d}p_{M}\left(w,d\right)
\]
\end_inset
\end_layout
\begin_layout Standard
The above is all fine and everything, but inappropriate for comparing two
words.
The inner product between two words
\begin_inset Formula $w,u$
\end_inset
is
\begin_inset Formula
\[
i\left(w,u\right)=\sum_{d}N\left(w,d\right)N\left(u,d\right)
\]
\end_inset
and can be written as a proper (joint) probability as
\begin_inset Formula
\[
p\left(w,u\right)=\frac{i\left(w,u\right)}{i\left(*,*\right)}=\sum_{d}p_{M}\left(w,d\right)p_{M}\left(u,d\right)
\]
\end_inset
Note the absence of a subscript
\begin_inset Formula $M$
\end_inset
; this is a different probability than above.
\end_layout
\begin_layout Standard
The marginal probability is then
\begin_inset Formula
\[
p\left(w\right)\equiv p\left(w,*\right)=\sum_{d}p_{M}\left(w,d\right)p_{M}\left(*,d\right)
\]
\end_inset
\end_layout
\begin_layout Standard
The fractional MI between two words is then
\begin_inset Formula
\[
MI\left(w,u\right)=\log_{2}\frac{p\left(w,u\right)}{p\left(w\right)p\left(u\right)}
\]
\end_inset
and so is now manifestly compatible with Wikipedia.
\end_layout
\begin_layout Standard
This has the desirable property of giving the same score to synonyms.
That is, if
\begin_inset Formula $N\left(u,d\right)=\alpha N\left(w,d\right)$
\end_inset
for some constant
\begin_inset Formula $\alpha$
\end_inset
, then the two words
\begin_inset Formula $u$
\end_inset
and
\begin_inset Formula $w$
\end_inset
should be judged as perfect synonyms, and so should be perfectly similar.
Plugging through, we find that the MI obeys this condition:
\begin_inset Formula
\[
MI\left(u,u\right)=MI\left(w,w\right)=MI\left(u,w\right)
\]
\end_inset
when
\begin_inset Formula $N\left(u,d\right)=\alpha N\left(w,d\right)$
\end_inset
holds.
\end_layout
\begin_layout Standard
The (fractional) joint entropy is
\begin_inset Formula
\[
H\left(w,u\right)=-\log_{2}p\left(u,w\right)
\]
\end_inset
Note the minus sign.
\end_layout
\begin_layout Standard
Wikipedia then defines the
\begin_inset Quotes eld
\end_inset
variation of information
\begin_inset Quotes erd
\end_inset
\begin_inset Formula $d\left(u,w\right)$
\end_inset
as
\begin_inset Formula
\begin{align*}
d\left(u,w\right) & =H\left(u,w\right)-MI\left(u,w\right)\\
& =\log_{2}p\left(w\right)p\left(u\right)-2\log_{2}p\left(u,w\right)
\end{align*}
\end_inset
It then claims that
\begin_inset Formula
\[
0\le\sum_{w,u}p\left(w,u\right)d\left(w,u\right)\le\sum_{w,u}p\left(w,u\right)\log_{2}p\left(w,u\right)
\]
\end_inset
which is not manifestly obvious to me, but I guess it's plausible.
This suggests the
\begin_inset Quotes eld
\end_inset
normalized metric
\begin_inset Quotes erd
\end_inset
\begin_inset Formula
\[
D\left(w,u\right)=\frac{d\left(w,u\right)}{H\left(w,u\right)}=1-\frac{MI\left(w,u\right)}{H\left(w,u\right)}
\]
\end_inset
Is this really always positive? If so, its not superficially obvious to
me.
This is the fractional part of the
\begin_inset Quotes eld
\end_inset
Jaccard distance
\begin_inset Quotes erd
\end_inset
, arrived at when given the appropriate measure-theoretic interpretation.
There is also a different
\begin_inset Quotes eld
\end_inset
metric
\begin_inset Quotes erd
\end_inset
:
\begin_inset Formula
\[
D^{\prime}\left(w,u\right)=1-\frac{MI\left(w,u\right)}{\max\left(H\left(w\right),H\left(u\right)\right)}
\]
\end_inset
\end_layout
\begin_layout Subsubsection*
Jaccard MI
\end_layout
\begin_layout Standard
This section defines a kind-of
\begin_inset Quotes eld
\end_inset
Jaccard mutual information
\begin_inset Quotes erd
\end_inset
, in a kind of
\emph on
ad hoc
\emph default
manner, by inspiration.
It's not particularly well-founded; rather, its plausible.
Except that it's not – it's broken.
\end_layout
\begin_layout Standard
Define the support of a vector as
\begin_inset Formula
\[
S\left(w\right)=\left\{ d:N\left(w,d\right)>0\right\}
\]
\end_inset
so that the Jaccard union is
\begin_inset Formula
\[
S\left(w\right)\cup S\left(u\right)
\]
\end_inset
How about the rest, then? Well, lets just try it out.
Abusing the notation for conditional probability, write
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align*}
p\left(w;u,d\right) & =\frac{N\left(w,d\right)}{\sum_{d}N\left(w,d\right)+N\left(u,d\right)}\\
& =\frac{N\left(w,d\right)}{N\left(w,*\right)+N\left(u,*\right)}
\end{align*}
\end_inset
Then define a Jaccard MI as
\begin_inset Formula
\[
MI_{J}\left(w,u\right)=\log_{2}\frac{\left(\sum_{d}p(w;u,d)p(u;w,d)\right)\left(\sum_{d}p(*;u,d)p(*;w,d)\right)}{\left(\sum_{d}p(w;u,d)p(*;u,d)\right)\left(\sum_{d}p(u;w,d)p(*;w,d)\right)}
\]
\end_inset
Plugging through, this reduces to
\begin_inset Formula
\[
MI_{J}\left(w,u\right)=\log_{2}\frac{\left(\sum_{d}N(w,d)N(u,d)\right)\left(\sum_{d}p(*;u,d)p(*;w,d)\right)}{\left(\sum_{d}N(w,d)p(*;u,d)\right)\left(\sum_{d}N(u,d)p(*;w,d)\right)}
\]
\end_inset
where
\begin_inset Formula
\[
p(*;w,d)=\sum_{x}\frac{N\left(x,d\right)}{N\left(x,*\right)+N\left(w,*\right)}
\]
\end_inset
which is computationally awkward.
\end_layout
\begin_layout Standard
Upshot is: the conventional dot product is not altered.
The computation of this Jaccard MI is far more complex and time consuming.
Curiously, the obvious attempt to simplify/approximate it just takes us
back to the conventional MI.
\end_layout
\begin_layout Standard
The primary problem here is that this violates one of the desired properties
for the similarity of synonyms.
That is, if
\begin_inset Formula $N\left(u,d\right)=\alpha N\left(w,d\right)$
\end_inset
for some constant
\begin_inset Formula $\alpha$
\end_inset
, then the two words
\begin_inset Formula $u$
\end_inset
and
\begin_inset Formula $w$
\end_inset
should be judged as perfect synonyms, and the resulting similarity should
be perfect.
This distance violates that.
\end_layout
\begin_layout Section*
July 2021 - Similarity Smackdown
\end_layout
\begin_layout Standard
So, above, we have a whole variety of different similarities.
Lets see how they perform on real data.
This test is to be done on several datasets: `
\family sans
run-1-en_mpg-tranche-123.rdb
\family default
` and trimmed versions of this.
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
The `run-1-en_mpg-tranche-123.rdb` dataset contains 113M atoms.
Of these, 391548 (391K) are WordNodes
\end_layout
\end_inset
Its our midsize English dataset.
It's pretty big.
The comparison will be between the top ten most frequent common nouns,
the top ten verbs, the top ten adjectives, the top ten pronouns, the top
ten prepositions and the top ten adverbs.
The min, max, mean and rms similarity will be computed in-group, for each
group.
This will be between the 10*9/2=45 pairs in each group.
Next, the min, max, avg and rms similarity between groups will be computed;
this will run over 10*10=100 between-group pairs.
This will be repeated for the trimmed dataset, too.
(Maybe several trims?) Should probably do this with and without shapes.
This is a pretty massive experiment, with a whole lot of compute going
into it.
This is going to take a while, and spew a lot of data.
\end_layout
\begin_layout Standard
Lets go.
The word categories (these are hand-picked):
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
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POS
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
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Difficult set
\end_layout
\end_inset
</cell>
</row>
<row>
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\begin_inset Text
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determiner
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
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the a that this an some any these those most -
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
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preposition
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
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of to in with for at on by from up -
\end_layout
\end_inset
</cell>
</row>
<row>
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\begin_inset Text
\begin_layout Plain Layout
verb-ir
\end_layout
\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
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was is had be have were are would been will -
\end_layout
\end_inset
</cell>
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\begin_inset Text
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verb-act
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
said like do did know made see go came come -
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
pronoun
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
I he it you him she they me them we -
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
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possessive
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
his her my their your its our ones +
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
adverb
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
that as not no there about only then very just -
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
conjunction
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
and but or so if than both either yet nor -
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
adjective
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
all one other little even good great long old many -
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
WH-words
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
which who when what where why how while whatever whose -
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
noun
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
time man way eyes day hand head men face place -
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
aux
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
can should could may must might shall cannot am has -
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
Note that some of these are ambiguous:
\begin_inset Quotes eld
\end_inset
as
\begin_inset Quotes erd
\end_inset
can be an adverb or preposition.
\begin_inset Quotes eld
\end_inset
that
\begin_inset Quotes erd
\end_inset
can be an adverb or a determiner.
As such, they might be excessively broad categories, and might overlap
a fair bit.
\end_layout
\begin_layout Standard
The table below defines an
\begin_inset Quotes eld
\end_inset
easier
\begin_inset Quotes erd
\end_inset
set, that is, hand-picked, because they seem easier to me.
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="18" columns="2">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
POS
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Easier set
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
determiner
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
the a this an some any these those all most -
\end_layout
\end_inset
</cell>
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<row>
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\begin_inset Text
\begin_layout Plain Layout
preposition
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
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in with for at on by from up out into -
\end_layout
\end_inset
</cell>
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<row>
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\begin_inset Text
\begin_layout Plain Layout
verb-ir
\end_layout
\end_inset
</cell>
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\begin_inset Text
\begin_layout Plain Layout
was is be have were are been has having being -
\end_layout
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</cell>
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<row>
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\begin_inset Text
\begin_layout Plain Layout
verb-act
\end_layout
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\begin_inset Text
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like do know see go come make say think want -
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verb-past
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
said did thought looked found took asked saw told knew -
\end_layout
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</cell>
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<row>
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\begin_inset Text
\begin_layout Plain Layout
pronoun
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
I he it you him she they me them we -
\end_layout
\end_inset
</cell>
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<row>
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\begin_inset Text
\begin_layout Plain Layout
possessive
\end_layout
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\begin_layout Plain Layout
his her my their your its our ones +
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\begin_inset Text
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adverb
\end_layout
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\begin_inset Text
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not no there about only then very just rather often -
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conjunction
\end_layout
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\begin_inset Text
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and but or so if than both either yet nor -
\end_layout
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</cell>
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<row>
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adjective
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
one little good great long old many young black hard -
\end_layout
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</cell>
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<row>
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\begin_inset Text
\begin_layout Plain Layout
WH-words
\end_layout
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</cell>
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\begin_inset Text
\begin_layout Plain Layout
which who when what where why how while whatever whose -
\end_layout
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</cell>
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<row>
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\begin_inset Text
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comparative
\end_layout
\end_inset
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\begin_inset Text
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better longer greater lower higher closer earlier larger older younger -
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</cell>
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<row>
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\begin_layout Plain Layout
noun
\end_layout
\end_inset
</cell>
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\begin_layout Standard
A better selection technique might be to start with the
\begin_inset Quotes eld
\end_inset
difficult set
\begin_inset Quotes erd
\end_inset
and kick out the two outliers (according to conventional MI) and obtain
an easier set that way.
\end_layout
\begin_layout Standard
Anyway, the experiments below were all performed with the difficult set.
\end_layout
\begin_layout Subsection*
Summary of Results
\end_layout
\begin_layout Standard
The sections below provide detailed results.
To summarize, these are:
\end_layout
\begin_layout Itemize
MI beats the pants off of cosine similarity; of all the distance measures,
cosine similarity is in a distant last place.
\end_layout
\begin_layout Itemize
Surprisingly, three different metrics provide superior results to MI: overlap
similarity (unweighted Jaccard similarity), weighted Jaccard similarity
and
\begin_inset Quotes eld
\end_inset
probability
\begin_inset Quotes erd
\end_inset
Jaccard similarity, in order of quality of results.
These improve the inter-intra-cluster separation by over 10% compared to
using MI.
(So - 10% is not a lot, but we'll take what we can.)
\end_layout
\begin_layout Itemize
This last result is ...
surprising, and seems to contradict the naive belief in information theory.
How is this to be explained? Is there some deeper information-theoretic
explanation that has not been grasped?
\end_layout
\begin_layout Itemize
The experiment fails to measure the thing that originally inspired it: There
are word pairs with very high MI but very low overlap similarity, simply
because one of the word vectors is very short.
The hand-picked clusters above contain no short vectors!
\end_layout
\begin_layout Itemize
Surprisingly, the use of shapes makes the MI scores considerably worse.
It's possible, but not likely, that this is due to dataset filtering problems.
That experiment still needs to be performed.
Naive imagination suggests shapes should have made everything better.
\end_layout
\begin_layout Itemize
Shapes do seem to make things better, when using the various forms of the
Jaccard distance.
This is especially the case for the smaller/smallest datasets.
\end_layout
\begin_layout Standard
See also a different summary of results at the bottom, and another at the
top.
\end_layout
\begin_layout Subsection*
Trimmed dataset (Poorly trimmed)
\end_layout
\begin_layout Standard
Above is to be run on the following datasets.
Attention: The table immediately below summarizes datasets generated with
a poorly designed filtering operation.
The problem is that the `add-zero-filter` was not run, resulting in words
in Connectors that do not appear in Sections.
This not only alters product computations, but it also leads to cross-sections
whose head word doesn't appear in a section.
This turns out to skew results, but not all that much - mostly less than
a few percent.
So these datasets appear to be
\begin_inset Quotes eld
\end_inset
pretty good
\begin_inset Quotes erd
\end_inset
, but not entirely fully self-consistent.
See further below for the revised results.
Funny thing: those revised results are also incompletely self-consistent.
They're better, but it turns out a single pass of `add-zero-filter` is
not enough.
A recursive multi-pass will be required.
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\begin_layout Standard
The first two datasets do not have all of the words for all of the clusters.
The
\begin_inset Quotes eld
\end_inset
marg
\begin_inset Quotes erd
\end_inset
datasets use the Sections (pseudo-csets) only; the
\begin_inset Quotes eld
\end_inset
shape
\begin_inset Quotes erd
\end_inset
datasets include both Sections and Cross-Sections.
The number of words differ between the corresponding datasets, because
the filtering cut things in such a way that there are words appearing in
CrossSections that don't have a matching Section (because that Section
was cut by the filter).
That is, there are words in connectors that don't appear in Sections.
\end_layout
\begin_layout Standard
XXX The above should not have happened; `add-linkage-filter` should have
prevented the above.
Is there a bug? What's up with that? Answer: above failed to run `add-zero-filt
er` after the row-column filter, but before `add-linkage-filter`.
The problem was that `add-linkage-filter` was given a list of words that
had zero counts, because the generic filter does not knock out rows whose
counts are empty.
All this is recomputed further below.
\end_layout
\begin_layout Subsection*
r3-filt-640-128-80-marg.rdb – MI (for the poorly-filtered dataset)
\end_layout
\begin_layout Standard
So lets try it out.
Attention: this is done with the poorly-filtered datasets above, but apparently
can still be taken to be
\begin_inset Quotes eld
\end_inset
representative
\begin_inset Quotes erd
\end_inset
.
Better filtering changes results by only a few percent, or less.
\end_layout
\begin_layout Standard
The r3-filt-640-128-80-marg.rdb dataset is the first dataset big enough to
measure.
The stats are for the MI.
The cluster stats are:
\end_layout
\begin_layout Standard
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\begin_layout Standard
This dataset is small enough that not all of the words in the cluster appear.
The `sz` column counts how many of those words are present.
The
\begin_inset Formula $-\infty$
\end_inset
in the `min` column shows that this dataset has word-pairs with zero overlap.
Yow! The `pr` column counts how many good pairs there were (there should
be 49=10x9/2, if they were all comparable).
The average and rms exclude such pairs.
\end_layout
\begin_layout Standard
Conclusion: this dataset is not big enough to sample many common English
words.
It is big enough to classify at least some words into clusters, though.
\end_layout
\begin_layout Standard
How about between clusters? Worth a quick look.
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5-pronoun 6-posessive
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7-adverb 8-conj
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either .
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\end_inset
\end_layout
\begin_layout Standard
There are 66=12x11/2 of these cluster pairs.
The above samples just some of them.
it seems to show some problems with these hand-picked clusters: they overlap
more than imagined, and not surprisingly so.
The only real surprise here is the
\begin_inset Quotes eld
\end_inset
see-him
\begin_inset Quotes erd
\end_inset
pair.
\end_layout
\begin_layout Subsection*
r3-filt-40-8-5-marg.rdb – MI (for the poorly-filtered dataset)
\end_layout
\begin_layout Standard
A much bigger dataset than the above.
Attention: this is done with the poorly-filtered dataset described above.
The results seem to still be
\begin_inset Quotes eld
\end_inset
representative
\begin_inset Quotes erd
\end_inset
; better filtering changes them by a few percent, sometimes less.
\end_layout
\begin_layout Standard
The stats are for the MI.
The cluster stats are:
\end_layout
\begin_layout Standard
\align center
\begin_inset VSpace defskip
\end_inset
\begin_inset Tabular
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<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
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<column alignment="center" valignment="top">
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\begin_layout Standard
The green colors are the words that are the same as in the 640-128-80 dataset.
Many means and most rms are similar to the smaller dataset.
So that's good: the datasets are reflective of one-another.
The means are ..
low, but perhaps not surprisingly low, as they are a rather disparate collectio
n of words...
\end_layout
\begin_layout Standard
Rather than reporting on all 66 inter-cluster distances, it seems simplest
to provide a single score: the grand-total average and rms of the inter-cluster
scores.
There are 4617 inter-cluster pairs, with a mean of -1.32 and rms of 3.007.
Comparing to the intra-cluster mean and rms, it looks like the difference
is 3.70, or about 1.3 std-dev distance between the two.
\end_layout
\begin_layout Subsection*
Inter-intra MI summary (for the poorly-filtered datasets)
\end_layout
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Pursuing above, the table below records the inter and intra cluster average
MI for the datasets.
\end_layout
\begin_layout Standard
Attention: The table below is generated with the poorly-filtered datasets
described above.
A table further below presents the same data for better-filtered datasets.
Comparing tables, one sees they are mostly the same: most numbers change
by a few percent; some differences are as large as 10% and some are under
one percent.
So this is mostly representative.
\end_layout
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The big question here is why shapes make things worse; this remains a big
unresolved question.
\end_layout
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\begin_layout Standard
The
\begin_inset Quotes eld
\end_inset
separation
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\end_inset
is the number of stddev between the intra and inter MI values.
\end_layout
\begin_layout Subsection*
Trimmed dataset
\end_layout
\begin_layout Standard
Redo of trimming, as explained above.
These are more tightly trimmed, but still not fully trimmed.
See below for discussion.
Time to perform the filtering is about 4200 seconds for each file.
\end_layout
\begin_layout Standard
Notation: filename-WW-DD-SS-foo where WW is the cutoff for the minimum time
this word has been observed.
DD is the cutoff for the number of times a disjunct has been observed,
and SS is the cutoff for the number of times a given section has been observed.
This is in keeping with how the filter object is designed in the main atomspace
repo – these are the three filter parameters.
\end_layout
\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
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\strikeout off
\xout off
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\noun off
\color none
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\end_inset
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\end_inset
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\begin_layout Standard
Some of the thinner datasets do not have all of the words for all of the
clusters.
The
\begin_inset Quotes eld
\end_inset
marg
\begin_inset Quotes erd
\end_inset
datasets use the Sections (pseudo-csets) only; the
\begin_inset Quotes eld
\end_inset
shape
\begin_inset Quotes erd
\end_inset
datasets include both Sections and Cross-Sections.
\end_layout
\begin_layout Standard
So here's the trimming problem: the algo used is not yet fully recursive.
So: first round of trimming removes sections which have connectors that
don't appear as words in the left-basis.
This leaves some words with no sections on them.
These are removed.
However, those words might still appear in some connectors.
Thus a second round of trimming is needed.
And so on, and so on, until a stable collection is arrived at.
\end_layout
\begin_layout Standard
Some additional confusion: there might be words in the left basis that never
appear in connectors.
These also have to be removed.
When these are removed, corresponding sections disappear as well, necessitating
another recursive round.
Yech.
So lets proceed without this, and see what happens.
We can try again, later.
\end_layout
\begin_layout Standard
Notes: `
\family sans
r3-mpg-marg.rdb
\family default
` is just a copy of `
\family sans
run-1-en_mpg-tranche-123.rdb
\family default
` with MMT marginals for Sections only (no CrossSections).
\end_layout
\begin_layout Subsection*
Inter-intra MI summary
\end_layout
\begin_layout Standard
The table below records the inter and intra cluster average MI for the datasets.
\end_layout
\begin_layout Standard
This proceeds as follows.
Given one cluster, the MI was computed for each pair of words in the cluster.
These were then averaged together to obtain the average cluster MI, and
also the RMS for that cluster.
The averaging is unweighted (all word-pairs get a uniform weight).
Next, all of these cluster scores were averaged together, to give the
\begin_inset Quotes eld
\end_inset
intra MI
\begin_inset Quotes erd
\end_inset
for the entire dataset.
This second averaging was weighted by the number of pairs in each cluster;
the result is that all word-pairs get a uniform weight.
The
\begin_inset Quotes eld
\end_inset
tot
\begin_inset Quotes erd
\end_inset
column records the total number of pairs.
In formulas:
\begin_inset Formula
\[
intra=\sum_{c\in C}\;\sum_{w,u\in c}MI(u,w)
\]
\end_inset
where
\begin_inset Formula $C$
\end_inset
is the set of all of the clusters.
\end_layout
\begin_layout Standard
The
\begin_inset Quotes eld
\end_inset
inter MI
\begin_inset Quotes erd
\end_inset
column computes the average MI over all word-pairs, where the two words
are in distinct clusters.
That is, each word is in a different cluster:
\begin_inset Formula
\[
inter=\sum_{c\in C}\;\sum_{c^{\prime}\in C;c^{\prime}\ne c}\;\sum_{w\in c;u\in c^{\prime}}MI(u,w)
\]
\end_inset
The
\begin_inset Quotes eld
\end_inset
tot
\begin_inset Quotes erd
\end_inset
column records the total number of such pairs.
\end_layout
\begin_layout Standard
The
\begin_inset Quotes eld
\end_inset
separation
\begin_inset Quotes erd
\end_inset
is the number of stddev's between the intra and inter MI values.
The stddev is the larger of the two RMS's.
\end_layout
\begin_layout Standard
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\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
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</lyxtabular>
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\align center
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\end_inset
\end_layout
\begin_layout Standard
So we conclude:
\end_layout
\begin_layout Itemize
Less filtering gives better results.
\end_layout
\begin_layout Itemize
Shapes make all vectors more similar to each other.
The RMS with shapes is smaller, considerably smaller, with shapes, than
without them.
\end_layout
\begin_layout Itemize
Unfortunately, this is true not only within clusters, but also between them!
Shapes fail to improve the discrimination between clusters.
This is counter-intuitive.
\end_layout
\begin_layout Itemize
Shapes are stunningly disappointing; the reason for this is quite unclear.
It may be due to the faulty filtering, and so deserves one more round.
Yet there's a hint that even better filtering still won't improve things.
So this is an open problem.
\end_layout
\begin_layout Subsection*
Inter-intra joint probability summary
\end_layout
\begin_layout Standard
The table below records the inter and intra cluster average joint probability
for the datasets.
The joint probability is as defined previously; it is the normalized inner
product.
\begin_inset Formula
\[
p\left(w,u\right)=\frac{i\left(w,u\right)}{i\left(*,*\right)}=\sum_{d}p_{M}\left(w,d\right)p_{M}\left(u,d\right)
\]
\end_inset
with
\begin_inset Formula
\[
p_{M}\left(w,d\right)=\frac{N\left(w,d\right)}{N\left(*,*\right)}
\]
\end_inset
\end_layout
\begin_layout Standard
Several problems arise when computing the
\begin_inset Quotes eld
\end_inset
average probability
\begin_inset Quotes erd
\end_inset
for words in a cluster or between clusters.
Some words are observed far more frequently than others, and we really
do want to adjust for that.
Thus, it is best to compare the joint probability divided by the marginal
probabilities for each word:
\begin_inset Formula
\[
\frac{p\left(w,u\right)}{p_{M}\left(w,*\right)p_{M}\left(u,*\right)}
\]
\end_inset
where the wild-cards range over the disjuncts.
\end_layout
\begin_layout Standard
Next, computing the RMS for the above doesn't really make sense, cause probabili
ties cannot go negative.
It seems better to compute the averages and RMS for
\begin_inset Formula $\log_{2}p$
\end_inset
, that is, for
\begin_inset Formula
\[
\log_{2}\frac{p\left(w,u\right)}{p_{M}\left(w,*\right)p_{M}\left(u,*\right)}
\]
\end_inset
This should be quite similar to the MI.
\end_layout
\begin_layout Standard
Here,
\begin_inset Quotes eld
\end_inset
larger is better
\begin_inset Quotes erd
\end_inset
– larger log2 numbers mean a better match.
\end_layout
\begin_layout Standard
\align center
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\end_inset
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\begin_layout Standard
\align center
\begin_inset Tabular
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\align center
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\end_layout
\begin_layout Standard
So we conclude:
\end_layout
\begin_layout Itemize
Less filtering gives better results.
\end_layout
\begin_layout Itemize
Quality is close to the MI results, and only slightly poorer.
\end_layout
\begin_layout Itemize
Shapes offer no advantage; they're a clear disadvantage.
\end_layout
\begin_layout Subsection*
Inter-intra variation of information summary
\end_layout
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The table below records the inter and intra cluster average
\begin_inset Quotes eld
\end_inset
variation of information
\begin_inset Quotes erd
\end_inset
for the datasets.
This is defined as
\begin_inset Formula
\[
VI\left(u,w\right)=H\left(u,w\right)-MI\left(u,w\right)
\]
\end_inset
where
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\end_inset
is the log of the joint probability, as defined before:
\end_layout
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\begin_inset Formula
\[
H\left(w,u\right)=-\log_{2}p\left(w,u\right)
\]
\end_inset
\end_layout
\begin_layout Standard
Here,
\begin_inset Quotes eld
\end_inset
smaller is better
\begin_inset Quotes erd
\end_inset
– smaller numbers mean a better match.
Per Wikipedia, this is a metric (obeying the triangle inequality).
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\align center
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\end_inset
\end_layout
\begin_layout Standard
\align center
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\begin_layout Standard
So we conclude:
\end_layout
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A touch better than the joint-probability results above.
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Separation is not as good as MI, but are within 5% of the MI separation.
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Inter-intra normalized variation of information summary
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The table below records the inter and intra cluster average
\begin_inset Quotes eld
\end_inset
normalized variation of information
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\end_inset
for the datasets.
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\begin_inset Formula
\[
NVI\left(u,w\right)=\frac{MI\left(u,w\right)}{H\left(u,w\right)}
\]
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larger is better
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– larger numbers mean a better match.
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So we conclude:
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Separation is a fair bit better than the un-normalized variation of information.
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Separation is about 10% better than the MI separation, for the plain sections.
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Separation is about 5% better than the MI separation, for the shapes.
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Inter-intra cosine summary
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The table below records the inter and intra cluster average cosine distance
for the datasets.
Although this is
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obviously
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inappropriate for this work, it is the industry standard benchmark, so
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So we conclude:
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This is not competitive with the probability-based methods, not even close.
\end_layout
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Shapes improve the cosine distance! Even though this is still with non-optimal
filtering! Hypothesis: the shape vectors really do improve vector alignment
(as this clearly shows) but the extra MI terms dissolve it back out.
That is, the shapes do not carry any extra information not already present
in the vectors.
\end_layout
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This may be a misinterpretation, though.
By comparing inter scores with and without shapes, we see that the shapes
make the vectors more similar.
This is also the case for the intra scores.
In particular, the RMS for the intra scores goes down (cause they're more
similar) which makes the separation improve.
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Conclude: shapes make all vectors appear to be more similar.
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\begin_layout Subsection*
Inter-intra log cosine summary
\end_layout
\begin_layout Standard
The table below records the inter and intra cluster average log2 cosine
distance for the datasets.
This is similar to above, except that the log2 makes the numbers more directly
comparable to the probabilities.
The cosine ranges from 0.0 to 1.0, with 1.0 meaning
\begin_inset Quotes eld
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highly similar
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.
Thus, the log emphasizes dissimilarities.
Larger values are better.
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Doing this does alter the contributions to the averages: for the smaller
datasets, zero cosines are frequent.
This is due to the dataset size: just not enough disjuncts.
These do contribute to the average cosine distance, but not to the log
distance; they're discarded (since log zero is minus infinity).
This seems to be an OK thing to do for the present situation.
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\begin_layout Standard
So we conclude:
\end_layout
\begin_layout Itemize
(Looking at the `separation` column:) This is competitive with the joint-probabi
lity scores, maybe even a hair better, and is just a little worse (about
8% worse) than the MI scores.
\end_layout
\begin_layout Itemize
Unlike the plain cosine, the shapes make things worse.
The log2 has the effect of magnifying differences (since the log is almost
linear for high similarity, but stretches the scale strongly for low similarity.
) And yet, if we compare the inter scores with and without shapes, it seems
that shapes are making vectors more similar.
This is consistent with the non-log cosine, and the separation is controlled
by the RMS.
\end_layout
\begin_layout Subsection*
Inter-intra overlap summary
\end_layout
\begin_layout Standard
The table below records the inter and intra cluster average overlap similarity
for the datasets.
This is provided per API for the `add-similarity-compute` object.
The overlap is simply the unweighted Jaccard similarity.
This ranges from 0.0 to 1.0, with the interpretation that 1.0 means very similar.
\end_layout
\begin_layout Standard
That is, define the indicator function as
\begin_inset Formula
\[
\sigma\left(w,d\right)=\left[N\left(w,d\right)>0\right]
\]
\end_inset
Then the overlap similarity is
\begin_inset Formula
\begin{align*}
\mbox{overlap}\left(w,u\right)= & \sum_{d}\frac{\min\left(\sigma\left(w,d\right),\sigma\left(u,d\right)\right)}{\max\left(\sigma\left(w,d\right),\sigma\left(u,d\right)\right)}\\
= & \frac{\left|S\left(w\right)\cap S\left(u\right)\right|}{\left|S\left(w\right)\cup S\left(u\right)\right|}
\end{align*}
\end_inset
where the support of a vector is defined as
\begin_inset Formula
\[
S\left(w\right)=\left\{ d:N\left(w,d\right)>0\right\}
\]
\end_inset
\end_layout
\begin_layout Standard
The averages and RMS are computed for
\begin_inset Formula $\log_{2}\mbox{overlap}\left(w,u\right)$
\end_inset
.
This is done for two reasons.
One is that it makes the reporting scale similar to that for the probability,
above.
The log is approx linear for high similarity, and thus emphasizes differences
for low similarity.
As before,
\begin_inset Quotes eld
\end_inset
larger is better
\begin_inset Quotes erd
\end_inset
– larger log2 numbers mean a better match.
\end_layout
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\align center
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\end_inset
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\begin_layout Standard
\align center
\begin_inset Tabular
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\end_inset
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\begin_layout Standard
\align center
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
So we conclude:
\end_layout
\begin_layout Itemize
For the shapeless sections, this provides superior separation to the MI
scores.
Its about 10% or 15% better.
\end_layout
\begin_layout Itemize
With shapes, the separation is even better – about 15% to 40% better than
the MI scores.
\end_layout
\begin_layout Itemize
The separation with shapes is mostly better than that without, so this breaks
a trend.
\end_layout
\begin_layout Itemize
Shapes give the most improvement for the small datasets! They're not useful,
once the dataset gets large.
\end_layout
\begin_layout Itemize
However, as before, the addition of shapes seems to make vectors more similar,
in general, no matter what kind they are.
\end_layout
\begin_layout Subsection*
Inter-intra Jaccard summary
\end_layout
\begin_layout Standard
The table below records the inter and intra cluster average
\begin_inset Quotes eld
\end_inset
conditional Jaccard similarity
\begin_inset Quotes erd
\end_inset
for the datasets.
This is provided by the `cond-jacc` method on the `add-similarity-compute`
object.
This is the same as that defined, using above: its the Ruzicka similarity
for the conditional probabilities.
To recap, so that I don't have to scroll up and re-read, the conditional
Jaccard similarity is given by
\begin_inset Formula
\[
J_{R}\left(N;u,w\right)=\frac{\sum_{d}\min\left(p\left(w,d\right),p\left(u,d\right)\right)}{\sum_{d}\max\left(p\left(w,d\right),p\left(u,d\right)\right)}
\]
\end_inset
where
\begin_inset Formula $p\left(w,d\right)=N\left(w,d\right)/N\left(*,*\right)$
\end_inset
.
\end_layout
\begin_layout Standard
The conditional Jaccard similarity ranges from 0.0 to 1.0.
As above, log2 is taken.
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So we conclude:
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For the shapeless sections, this provides almost as good separation as the
overlaps, and about 5% better than the MI scores.
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Using shapes seems to improve separability by a fair amount – again, bucking
a trend.
Most of the improvement is in the smaller datasets.
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As before, using shapes seems to make all vectors more similar to one-another.
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Inter-intra Probability Jaccard summary
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The table below records the inter and intra cluster average
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probability Jaccard distance
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for the datasets.
This is provided by the
\family typewriter
`add-similarity-compute`
\family default
object, by calling the
\family typewriter
'right-prjacc
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\begin_layout Standard
\align center
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\end_layout
\begin_layout Standard
So we conclude:
\end_layout
\begin_layout Itemize
For the shapeless sections, this provides the best separation of all, and
is just a tad better than the conditional Jaccard, above.
\end_layout
\begin_layout Itemize
For shapeless separation, this is about 5% to 10% better than MI.
\end_layout
\begin_layout Itemize
Adding shapes further improves separability, again bucking a trend.
\end_layout
\begin_layout Standard
This is stunningly painfully slow for the larger datasets, and so the algo
needs to be refactored to move common computations out of the inner loop.
If this cannot be done, then this also is useless.
\end_layout
\begin_layout Subsection*
Summary
\end_layout
\begin_layout Standard
Lets try to summarize the results above.
The table below reproduces the separations for the various different measures.
The column titles are in the same order as in the text.
There are nine columns.
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\align center
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<column alignment="center" valignment="top">
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So we conclude:
\end_layout
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In all cases, the less that the dataset is trimmed, the greater the separation
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That is, trimming is unambiguously harmful to the quality of the data.
\end_layout
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Cosine distance is stunningly bad.
It's in a league of it's own.
Other than that, the scores are actually pretty similar, differing by at
most dozens of percent.
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The MI is a middle-of-the-road measure.
It is better than four columns: joint-probability, variation of information,
cosine, log cosine, but worse than the remaining four: normalized VI, overlap,
conditional Jaccard and probability Jaccard.
\end_layout
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Shapes have a mixed effect on the discrimination between clusters.
They make things much worse for the five worst metrics.
They make things better for the three best metrics, but only when the datasets
are heavily trimmed.
For the least-trimmed dataset, shapes make things worse.
\end_layout
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Ranking depends on the amount of trimming.
Different rankings result, depending on how much trimming one does.
The table below shows first through fifth places for rankings.
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</lyxtabular>
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\begin_layout Standard
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So we conclude:
\end_layout
\begin_layout Itemize
Overlap is a clear winner, across the board.
\end_layout
\begin_layout Itemize
Conditional Jaccard and Probability Jaccard are close to overlap, both for
the plain disjuncts, and for the shapes.
\end_layout
\begin_layout Itemize
For the shape datasets, 4th and 5th place are very distant from 3rd place.
\end_layout
\begin_layout Itemize
From first principles, one would have concluded that prjac should have done
best, as it is in some sense
\begin_inset Quotes eld
\end_inset
optimal
\begin_inset Quotes erd
\end_inset
.
Yet here, it rarely gets out of third place, hitting second place only
a few times.
The fact that it is very computationally expensive is a further strike
against it.
\end_layout
\begin_layout Itemize
Overlap beats the dominant 2nd place holder by a fair bit.
One upset: large shapes.
When overlap was in second place, it was by a slim margin.
\end_layout
\begin_layout Itemize
nvi makes a few showings, but does not seem to be worth considering any
further.
When its not in 1st place, its pretty distant from overlap.
\end_layout
\begin_layout Itemize
condj seems to be overall second place, and a fairly strong contender.
\end_layout
\begin_layout Standard
Not visible in the above table are the above factoids, observed elsewhere:
\end_layout
\begin_layout Itemize
prjac is very compute intensive.
\end_layout
\begin_layout Itemize
Shapes consistently reduce the differences between vectors.
Both inter-cluster differences are smaller but also intra-cluster differences
are also smaller.
Both RMS scores are pretty much always tighter, with shapes.
\end_layout
\begin_layout Itemize
Shapes have a mixed effect on discrimination between clusters.
Discussed in the paragraph above.
\end_layout
\begin_layout Itemize
Using shapes has a large impact on RAM and CPU usage.
It takes a lot of time to create them, they triple or quadruple RAM usage,
and all vector similarity computations take longer, because the vectors
are longer.
\end_layout
\begin_layout Itemize
Most of the tables have no results for the full dataset.
That's because computing with the full dataset is very CPU intensive.
We conclude that a very mild amount of trimming, to eliminate the onesie-twosie
counts is likely to be well worth the effort.
Particularly notable is that the scores for overlap and condj are tied
for the full dataset, and MI is in a close 3rd place.
Rerunning the experiment with more lightly trimmed datasets might be worthwhile.
(If rerun, then the
\begin_inset Quotes eld
\end_inset
better
\begin_inset Quotes erd
\end_inset
clusters should be used.
This will prevent direct comparison, but so what.)
\end_layout
\begin_layout Standard
To summarize: moving forward, it seems like we should eschew trimming, and
use overlap for the similarity measure.
However, understanding conditional Jaccard is also worthwhile.
Retaining MI as a benchmark also seems prudent, given how much effort has
been sunk into it.
If there's time, then understanding how the
\begin_inset Quotes eld
\end_inset
optimal
\begin_inset Quotes erd
\end_inset
probability Jaccard seems not optimal for this problem is worthwhile.
\end_layout
\begin_layout Standard
Apparently, the use of shapes is not worth the effort.
\end_layout
\begin_layout Section*
July 2021 - Big English
\end_layout
\begin_layout Standard
Attempt to do the big English data set again,
\begin_inset Quotes eld
\end_inset
correctly
\begin_inset Quotes erd
\end_inset
, this time.
The corpus comes in five parts, four from project Gutenberg, a fifth that
is fan fiction.
This naturally lends itself to datasets of increasing size.
Table below shows some stats.
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\end_layout
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So...
interestingly, the entropy is roughly independent of the dataset size.
Sparsity is high and increasing.
That's good.
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Table below is for MPG (Max Planar Graph) Parsing.
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6488s
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\end_inset
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<row>
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Wall-clock time (hours)
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\end_inset
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7.0 h
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4.3 h
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13.1
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11.6
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\end_layout
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</row>
<row>
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RAM to load pairs
\end_layout
\end_inset
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\end_layout
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</cell>
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\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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41.3
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67.8
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93.1
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<row>
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Highest RAM usage
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\end_inset
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30.2 GB
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47.9
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92.6
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\end_layout
\end_inset
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<row>
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CPU time (hours)
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52.1
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85.7
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</cell>
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180.3
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267
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\end_layout
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</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
Atoms in database
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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36M
\end_layout
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58M
\end_layout
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</cell>
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113M
\end_layout
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\begin_layout Plain Layout
156M
\end_layout
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</cell>
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\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
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\begin_layout Plain Layout
RocksDB file size
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
4.43 GB
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
6.95 GB
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
13.8 GB
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
18.8GB
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\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
Wall-clock time is time to process that one particular tranche.
Cumulative wall-clock is that, plus the sum of the earlier ones.
\end_layout
\begin_layout Standard
Loading speed appears to be about 1GB/minute, or 1M atoms/min or 17K atoms/sec.
\end_layout
\begin_layout Section*
The End
\end_layout
\begin_layout Standard
This is the end of the diary.
It spans the time period of February-August 2021.
The next part is Part Three.
\end_layout
\end_body
\end_document