| … | … | @@ -410,15 +410,15 @@ sentence |
| 410 | 410 | |
| 411 | 411 | \begin_layout Standard |
| 412 | 412 | The corpus is |
| 413 | | \begin_inset Formula $\mathcal{C}=\left\{ S_{i}\right\} $ |
| 413 | \begin_inset Formula $\mathcal{C}=\left\{ S_{a}\right\} $ |
| 414 | 414 | \end_inset |
| 415 | 415 | |
| 416 | 416 | which is a set of sentences |
| 417 | | \begin_inset Formula $S_{i}$ |
| 417 | \begin_inset Formula $S_{a}$ |
| 418 | 418 | \end_inset |
| 419 | 419 | |
| 420 | 420 | labelled with some index |
| 421 | | \begin_inset Formula $i$ |
| 421 | \begin_inset Formula $a$ |
| 422 | 422 | \end_inset |
| 423 | 423 | |
| 424 | 424 | . |
| … | … | @@ -434,13 +434,13 @@ The corpus is |
| 434 | 434 | The Hamiltonian is then |
| 435 | 435 | \begin_inset Formula |
| 436 | 436 | \[ |
| 437 | | \mathcal{H}=\sum_{S_{i}\in\mathcal{C}}H\left(S_{i}\right) |
| 437 | \mathcal{H}=\sum_{S_{a}\in\mathcal{C}}H\left(S_{a}\right) |
| 438 | 438 | \] |
| 439 | 439 | |
| 440 | 440 | \end_inset |
| 441 | 441 | |
| 442 | 442 | where the sum ranges over all sentences in the corpus, and |
| 443 | | \begin_inset Formula $H\left(S_{i}\right)$ |
| 443 | \begin_inset Formula $H\left(S_{a}\right)$ |
| 444 | 444 | \end_inset |
| 445 | 445 | |
| 446 | 446 | is the total cost of the parse (linkage) of that sentence. |
| … | … | @@ -452,6 +452,141 @@ And so here we meet our first conceptual problem: the linkage is not unique: |
| 452 | 452 | in principle, all possible linkages. |
| 453 | 453 | \end_layout |
| 454 | 454 | |
| 455 | \begin_layout Standard |
| 456 | So here we flop over the a graph/network representation. |
| 457 | First, lets drop the subscript |
| 458 | \begin_inset Formula $a$ |
| 459 | \end_inset |
| 460 | |
| 461 | on |
| 462 | \begin_inset Formula $S_{a}$ |
| 463 | \end_inset |
| 464 | |
| 465 | and just say |
| 466 | \begin_inset Formula $S\in\mathcal{C}$ |
| 467 | \end_inset |
| 468 | |
| 469 | is some sentence in the training corpus, and the Hamiltonian will be a |
| 470 | sum over these. |
| 471 | That doesn't change. |
| 472 | \end_layout |
| 473 | |
| 474 | \begin_layout Standard |
| 475 | Next, we say that each sentence |
| 476 | \begin_inset Formula $S=\left[w_{1},w_{2},\cdots,w_{n}\right]$ |
| 477 | \end_inset |
| 478 | |
| 479 | is a temporal sequence of specific words |
| 480 | \begin_inset Formula $w_{i}$ |
| 481 | \end_inset |
| 482 | |
| 483 | aka |
| 484 | \begin_inset Quotes eld |
| 485 | \end_inset |
| 486 | |
| 487 | tokens |
| 488 | \begin_inset Quotes erd |
| 489 | \end_inset |
| 490 | |
| 491 | in conventional machine learning. |
| 492 | Here, the index |
| 493 | \begin_inset Formula $i$ |
| 494 | \end_inset |
| 495 | |
| 496 | has a specific meaning: it indicates word–order. |
| 497 | Sequential time. |
| 498 | \end_layout |
| 499 | |
| 500 | \begin_layout Standard |
| 501 | For each sentence |
| 502 | \begin_inset Formula $S$ |
| 503 | \end_inset |
| 504 | |
| 505 | , consider the set |
| 506 | \begin_inset Formula $\mathcal{G}\left(S\right)=\left\{ G\left(S\right)\right\} $ |
| 507 | \end_inset |
| 508 | |
| 509 | of all possible connected graphs |
| 510 | \begin_inset Formula $G\left(S\right)$ |
| 511 | \end_inset |
| 512 | |
| 513 | for that sentence. |
| 514 | A graph here is a set of edges and vertexes: |
| 515 | \begin_inset Formula $G=\left\{ V,E\right\} $ |
| 516 | \end_inset |
| 517 | |
| 518 | (dropping the indicator |
| 519 | \begin_inset Formula $S$ |
| 520 | \end_inset |
| 521 | |
| 522 | because we fix the sentence for now.) For a given sentence |
| 523 | \begin_inset Formula $S$ |
| 524 | \end_inset |
| 525 | |
| 526 | , the set of vertexes is fixed: |
| 527 | \begin_inset Formula $V=\left\{ w_{i}|w_{i}\in S\right\} $ |
| 528 | \end_inset |
| 529 | |
| 530 | is the set of the words in the sentence |
| 531 | \begin_inset Formula $S$ |
| 532 | \end_inset |
| 533 | |
| 534 | . |
| 535 | As a set, the order no longer really matters. |
| 536 | But it is a multi-set, so its OK to have a word show repeatedly. |
| 537 | An edge is just a conventional graph edge: it is an ordered pair of vertexes. |
| 538 | Notation: |
| 539 | \begin_inset Formula $E_{ij}=\left(w_{i},w_{j}\right)$ |
| 540 | \end_inset |
| 541 | |
| 542 | is the edge connecting words at locations |
| 543 | \begin_inset Formula $i,j$ |
| 544 | \end_inset |
| 545 | |
| 546 | . |
| 547 | It is an ordered pair: edges have direction. |
| 548 | For a fixed graph |
| 549 | \begin_inset Formula $G$ |
| 550 | \end_inset |
| 551 | |
| 552 | the set of edges can be thought of that the adjacency matrix. |
| 553 | This is all very super-basic stuff, but I am trying to be as clear as posible |
| 554 | here. |
| 555 | Sorry. |
| 556 | The adjacency matrix is usually take to be a matrix woth just ones and |
| 557 | zeros in it, with one's taken to mean there is an edge, and zero meaning |
| 558 | there is not. |
| 559 | This can be generalized, but we're not doing that yet. |
| 560 | \end_layout |
| 561 | |
| 562 | \begin_layout Standard |
| 563 | A connected graph is one where every vertex is reachable by a path from |
| 564 | any other vertex. |
| 565 | We want to consider only connected graphs; the ability to disconnect will |
| 566 | be handled via a different machanism. |
| 567 | \end_layout |
| 568 | |
| 569 | \begin_layout Standard |
| 570 | For a fixed senttence |
| 571 | \begin_inset Formula $S$ |
| 572 | \end_inset |
| 573 | |
| 574 | , write the Hamiltonian |
| 575 | \begin_inset Formula |
| 576 | \[ |
| 577 | H\left(S\right)=\sum_{G\in\mathcal{G}}h\left(G\right) |
| 578 | \] |
| 579 | |
| 580 | \end_inset |
| 581 | |
| 582 | The sum ranges over all possible connected graphs over the words in the |
| 583 | sentence. |
| 584 | \end_layout |
| 585 | |
| 586 | \begin_layout Standard |
| 587 | Here, we arrive at the next stumbling block |
| 588 | \end_layout |
| 589 | |
| 455 | 590 | \begin_layout Section* |
| 456 | 591 | The End |
| 457 | 592 | \end_layout |