{ "paper_id": "P12-1024", "header": { "generated_with": "S2ORC 1.0.0", "date_generated": "2023-01-19T09:27:28.970868Z" }, "title": "Spectral Learning of Latent-Variable PCFGs", "authors": [ { "first": "Shay", "middle": [ "B" ], "last": "Cohen", "suffix": "", "affiliation": { "laboratory": "", "institution": "Columbia University", "location": {} }, "email": "scohen@cs.columbia.edu" }, { "first": "Karl", "middle": [], "last": "Stratos", "suffix": "", "affiliation": { "laboratory": "", "institution": "Columbia University", "location": {} }, "email": "stratos@cs.columbia.edu" }, { "first": "Michael", "middle": [], "last": "Collins", "suffix": "", "affiliation": { "laboratory": "", "institution": "Columbia University", "location": {} }, "email": "mcollins@cs.columbia.edu" }, { "first": "Dean", "middle": [ "P" ], "last": "Foster", "suffix": "", "affiliation": { "laboratory": "", "institution": "University of Pennsylvania", "location": {} }, "email": "foster@wharton.upenn.edu" }, { "first": "Lyle", "middle": [], "last": "Ungar", "suffix": "", "affiliation": {}, "email": "ungar@cis.upenn.edu" } ], "year": "", "venue": null, "identifiers": {}, "abstract": "We introduce a spectral learning algorithm for latent-variable PCFGs (Petrov et al., 2006). Under a separability (singular value) condition, we prove that the method provides consistent parameter estimates.", "pdf_parse": { "paper_id": "P12-1024", "_pdf_hash": "", "abstract": [ { "text": "We introduce a spectral learning algorithm for latent-variable PCFGs (Petrov et al., 2006). Under a separability (singular value) condition, we prove that the method provides consistent parameter estimates.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Abstract", "sec_num": null } ], "body_text": [ { "text": "Statistical models with hidden or latent variables are of great importance in natural language processing, speech, and many other fields. The EM algorithm is a remarkably successful method for parameter estimation within these models: it is simple, it is often relatively efficient, and it has well understood formal properties. It does, however, have a major limitation: it has no guarantee of finding the global optimum of the likelihood function. From a theoretical perspective, this means that the EM algorithm is not guaranteed to give consistent parameter estimates. From a practical perspective, problems with local optima can be difficult to deal with.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Recent work has introduced polynomial-time learning algorithms (and consistent estimation methods) for two important cases of hidden-variable models: Gaussian mixture models (Dasgupta, 1999; Vempala and Wang, 2004) and hidden Markov models (Hsu et al., 2009) . These algorithms use spectral methods: that is, algorithms based on eigenvector decompositions of linear systems, in particular singular value decomposition (SVD). In the general case, learning of HMMs or GMMs is intractable (e.g., see Terwijn, 2002) . Spectral methods finesse the problem of intractibility by assuming separability conditions. For example, the algorithm of Hsu et al. (2009) has a sample complexity that is polynomial in 1/\u03c3, where \u03c3 is the minimum singular value of an underlying decomposition. These methods are not susceptible to problems with local maxima, and give consistent parameter estimates.", "cite_spans": [ { "start": 174, "end": 190, "text": "(Dasgupta, 1999;", "ref_id": "BIBREF1" }, { "start": 191, "end": 214, "text": "Vempala and Wang, 2004)", "ref_id": "BIBREF13" }, { "start": 240, "end": 258, "text": "(Hsu et al., 2009)", "ref_id": "BIBREF4" }, { "start": 497, "end": 511, "text": "Terwijn, 2002)", "ref_id": "BIBREF12" }, { "start": 636, "end": 653, "text": "Hsu et al. (2009)", "ref_id": "BIBREF4" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "In this paper we derive a spectral algorithm for learning of latent-variable PCFGs (L-PCFGs) (Petrov et al., 2006; Matsuzaki et al., 2005) . Our method involves a significant extension of the techniques from Hsu et al. (2009) . L-PCFGs have been shown to be a very effective model for natural language parsing. Under a separation (singular value) condition, our algorithm provides consistent parameter estimates; this is in contrast with previous work, which has used the EM algorithm for parameter estimation, with the usual problems of local optima.", "cite_spans": [ { "start": 93, "end": 114, "text": "(Petrov et al., 2006;", "ref_id": "BIBREF10" }, { "start": 115, "end": 138, "text": "Matsuzaki et al., 2005)", "ref_id": "BIBREF7" }, { "start": 208, "end": 225, "text": "Hsu et al. (2009)", "ref_id": "BIBREF4" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "The parameter estimation algorithm (see figure 4) is simple and efficient. The first step is to take an SVD of the training examples, followed by a projection of the training examples down to a lowdimensional space. In a second step, empirical averages are calculated on the training example, followed by standard matrix operations. On test examples, simple (tensor-based) variants of the insideoutside algorithm (figures 2 and 3) can be used to calculate probabilities and marginals of interest.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Our method depends on the following results:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "\u2022 Tensor form of the inside-outside algorithm. Section 5 shows that the inside-outside algorithm for L-PCFGs can be written using tensors. Theorem 1 gives conditions under which the tensor form calculates inside and outside terms correctly.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "\u2022 Observable representations. Section 6 shows that under a singular-value condition, there is an observable form for the tensors required by the insideoutside algorithm. By an observable form, we follow the terminology of Hsu et al. (2009) in referring to quantities that can be estimated directly from data where values for latent variables are unobserved. Theorem 2 shows that tensors derived from the observable form satisfy the conditions of theorem 1.", "cite_spans": [ { "start": 222, "end": 239, "text": "Hsu et al. (2009)", "ref_id": "BIBREF4" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "\u2022 Estimating the model. Section 7 gives an algorithm for estimating parameters of the observable representation from training data. Theorem 3 gives a sample complexity result, showing that the estimates converge to the true distribution at a rate of 1/ \u221a M where M is the number of training examples.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "The algorithm is strikingly different from the EM algorithm for L-PCFGs, both in its basic form, and in its consistency guarantees. The techniques de-veloped in this paper are quite general, and should be relevant to the development of spectral methods for estimation in other models in NLP, for example alignment models for translation, synchronous PCFGs, and so on. The tensor form of the insideoutside algorithm gives a new view of basic calculations in PCFGs, and may itself lead to new models.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "For work on L-PCFGs using the EM algorithm, see Petrov et al. (2006) , Matsuzaki et al. (2005) , Pereira and Schabes (1992) . Our work builds on methods for learning of HMMs (Hsu et al., 2009; Foster et al., 2012; Jaeger, 2000) , but involves several extensions: in particular in the tensor form of the inside-outside algorithm, and observable representations for the tensor form. Balle et al. (2011) consider spectral learning of finite-state transducers; Lugue et al. (2012) considers spectral learning of head automata for dependency parsing. Parikh et al. (2011) consider spectral learning algorithms of treestructured directed bayes nets.", "cite_spans": [ { "start": 48, "end": 68, "text": "Petrov et al. (2006)", "ref_id": "BIBREF10" }, { "start": 71, "end": 94, "text": "Matsuzaki et al. (2005)", "ref_id": "BIBREF7" }, { "start": 97, "end": 123, "text": "Pereira and Schabes (1992)", "ref_id": "BIBREF9" }, { "start": 174, "end": 192, "text": "(Hsu et al., 2009;", "ref_id": "BIBREF4" }, { "start": 193, "end": 213, "text": "Foster et al., 2012;", "ref_id": "BIBREF2" }, { "start": 214, "end": 227, "text": "Jaeger, 2000)", "ref_id": "BIBREF5" }, { "start": 381, "end": 400, "text": "Balle et al. (2011)", "ref_id": "BIBREF0" }, { "start": 457, "end": 476, "text": "Lugue et al. (2012)", "ref_id": "BIBREF6" }, { "start": 546, "end": 566, "text": "Parikh et al. (2011)", "ref_id": "BIBREF8" } ], "ref_spans": [], "eq_spans": [], "section": "Related Work", "sec_num": "2" }, { "text": "Given a matrix A or a vector v, we write A \u22a4 or v \u22a4 for the associated transpose. For any integer n \u2265 1, we use [n] to denote the set {1, 2, . . . n}. For any row or column vector y \u2208 R m , we use diag(y) to refer to the (m \u00d7 m) matrix with diagonal elements equal to y h for h = 1 . . . m, and off-diagonal elements equal to 0. For any statement \u0393, we use [[\u0393] ] to refer to the indicator function that is 1 if \u0393 is true, and 0 if \u0393 is false. For a random variable X, we use E[X] to denote its expected value.", "cite_spans": [ { "start": 357, "end": 361, "text": "[[\u0393]", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Notation", "sec_num": "3" }, { "text": "We will make (quite limited) use of tensors:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Notation", "sec_num": "3" }, { "text": "Definition 1 A tensor C \u2208 R (m\u00d7m\u00d7m) is a set of m 3 parameters C i,j,k for i, j, k \u2208 [m]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Notation", "sec_num": "3" }, { "text": ". Given a tensor C, and a vector y \u2208 R m , we define C(y) to be the (m \u00d7 m) matrix with components", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Notation", "sec_num": "3" }, { "text": "[C(y)] i,j = k\u2208[m] C i,j,k y k . Hence C can be interpreted as a function C : R m \u2192 R (m\u00d7m) that maps a vector y \u2208 R m to a matrix C(y) of dimension (m \u00d7 m).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Notation", "sec_num": "3" }, { "text": "In addition, we define the tensor C * \u2208 R (m\u00d7m\u00d7m) for any tensor C \u2208 R (m\u00d7m\u00d7m) to have values", "cite_spans": [ { "start": 71, "end": 78, "text": "(m\u00d7m\u00d7m)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Notation", "sec_num": "3" }, { "text": "[C * ] i,j,k = C k,j,i Finally, for vectors x, y, z \u2208 R m , xy \u22a4 z \u22a4 is the tensor D \u2208 R m\u00d7m\u00d7m where D j,k,l = x j y k z l (", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Notation", "sec_num": "3" }, { "text": "this is analogous to the outer product:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Notation", "sec_num": "3" }, { "text": "[xy \u22a4 ] j,k = x j y k ).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Notation", "sec_num": "3" }, { "text": "This section gives a definition of the L-PCFG formalism used in this paper. An L-PCFG is a 5-tuple (N , I, P, m, n) where:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "\u2022 N is the set of non-terminal symbols in the grammar. I \u2282 N is a finite set of in-terminals. P \u2282 N is a finite set of pre-terminals. We assume that N = I \u222a P, and I \u2229 P = \u2205. Hence we have partitioned the set of non-terminals into two subsets.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "\u2022 [m] is the set of possible hidden states.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "\u2022 [n] is the set of possible words.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "\u2022 For all a \u2208 I, b \u2208 N , c \u2208 N , h 1 , h 2 , h 3 \u2208 [m], we have a context-free rule a(h 1 ) \u2192 b(h 2 ) c(h 3 ). \u2022 For all a \u2208 P, h \u2208 [m], x \u2208 [n]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": ", we have a context-free rule a(h) \u2192 x.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "Hence each in-terminal a \u2208 I is always the lefthand-side of a binary rule a \u2192 b c; and each preterminal a \u2208 P is always the left-hand-side of a rule a \u2192 x. Assuming that the non-terminals in the grammar can be partitioned this way is relatively benign, and makes the estimation problem cleaner.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "We define the set of possible \"skeletal rules\" as", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "R = {a \u2192 b c : a \u2208 I, b \u2208 N , c \u2208 N }.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "The parameters of the model are as follows:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "\u2022 For each a \u2192 b c \u2208 R, and h \u2208 [m], we have a parameter q(a \u2192 b c|h, a). For each a \u2208 P, These definitions give a PCFG, with rule probabilities", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "p(a(h 1 ) \u2192 b(h 2 ) c(h 3 )|a(h 1 )) = q(a \u2192 b c|h 1 , a) \u00d7 s(h 2 |h 1 , a \u2192 b c) \u00d7 t(h 3 |h 1 , a \u2192 b c) and p(a(h) \u2192 x|a(h)) = q(a \u2192 x|h, a).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "In addition, for each a \u2208 I, for each h \u2208 [m], we have a parameter \u03c0(a, h) which is the probability of non-terminal a paired with hidden variable h being at the root of the tree.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "An L-PCFG defines a distribution over parse trees as follows. A skeletal tree (s-tree) is a sequence of rules r 1 . . . r N where each r i is either of the form a \u2192 b c or a \u2192 x. The rule sequence forms a top-down, left-most derivation under a CFG with skeletal rules. See figure 1 for an example.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "A full tree consists of an s-tree Define a i to be the non-terminal on the left-handside of rule r i . For any i \u2208 {2 . . . N } define pa(i) to be the index of the rule above node i in the tree. Define L \u2282 [N ] to be the set of nodes in the tree which are the left-child of some parent, and R \u2282 [N ] to be the set of nodes which are the right-child of some parent. The probability mass function (PMF) over full trees is then", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "r 1 . . . r N , together with values h 1 . . . h N . Each h i is the value for S 1 NP 2 D 3 the N 4 dog VP 5 V 6 saw P 7 him r 1 = S \u2192 NP VP r 2 = NP \u2192 D N r 3 = D \u2192 the r 4 = N \u2192 dog r 5 = VP \u2192 V P r 6 = V \u2192 saw r 7 = P \u2192 him", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "p(r 1 . . . r N , h 1 . . . h N ) = \u03c0(a 1 , h 1 ) \u00d7 N i=1 q(r i |h i , a i ) \u00d7 i\u2208L s(h i |h pa(i) , r pa(i) ) \u00d7 i\u2208R t(h i |h pa(i) , r pa(i) )", "eq_num": "(1)" } ], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "The PMF over s-trees is", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "p(r 1 . . . r N ) = h 1 ...h N p(r 1 . . . r N , h 1 . . . h N ).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "In the remainder of this paper, we make use of matrix form of parameters of an L-PCFG, as follows:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "\u2022 For each a \u2192 b c \u2208 R, we define Q a\u2192b c \u2208 R m\u00d7m to be the matrix with values q(a \u2192 b c|h, a) for h = 1, 2, . . . m on its diagonal, and 0 values for its off-diagonal elements. Similarly, for each a \u2208 P, x \u2208 [n], we define Q a\u2192x \u2208 R m\u00d7m to be the matrix with values q(a \u2192 x|h, a) for h = 1, 2, . . . m on its diagonal, and 0 values for its off-diagonal elements.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "\u2022 For each a \u2192 b c \u2208 R, we define", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "S a\u2192b c \u2208 R m\u00d7m where [S a\u2192b c ] h \u2032 ,h = s(h \u2032 |h, a \u2192 b c). \u2022 For each a \u2192 b c \u2208 R, we define T a\u2192b c \u2208 R m\u00d7m where [T a\u2192b c ] h \u2032 ,h = t(h \u2032 |h, a \u2192 b c).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "\u2022 For each a \u2208 I, we define the vector \u03c0 a \u2208 R m where [\u03c0 a ] h = \u03c0(a, h).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "L-PCFGs: Basic Definitions", "sec_num": "4" }, { "text": "Given an L-PCFG, two calculations are central:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "Inputs: s-tree r1 . . . rN , L-PCFG (N , I, P, m, n), parameters \u2022 C a\u2192b c \u2208 R (m\u00d7m\u00d7m) for all a \u2192 b c \u2208 R \u2022 c \u221e a\u2192x \u2208 R (1\u00d7m) for all a \u2208 P, x \u2208 [n]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "\u2022 c 1 a \u2208 R (m\u00d71) for all a \u2208 I.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "Algorithm: (calculate the f i terms bottom-up in the tree)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "\u2022 For all i \u2208 [N ] such that ai \u2208 P, f i = c \u221e r i \u2022 For all i \u2208 [N ] such that ai \u2208 I, f i = f \u03b3 C r i (f \u03b2 )", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "where \u03b2 is the index of the left child of node i in the tree, and \u03b3 is the index of the right child.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "Return: 1. For a given s-tree", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "f 1 c 1 a 1 = p(r1 . . . rN )", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "r 1 . . . r N , calculate p(r 1 . . . r N ).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "2. For a given input sentence x = x 1 . . . x N , calculate the marginal probabilities", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "\u00b5(a, i, j) = \u03c4 \u2208T (x):(a,i,j)\u2208\u03c4 p(\u03c4 ) for each non-terminal a \u2208 N , for each (i, j) such that 1 \u2264 i \u2264 j \u2264 N .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "Here T (x) denotes the set of all possible s-trees for the sentence x, and we write (a, i, j) \u2208 \u03c4 if nonterminal a spans words x i . . . x j in the parse tree \u03c4 .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "The marginal probabilities have a number of uses. Perhaps most importantly, for a given sentence x = x 1 . . . x N , the parsing algorithm of Goodman (1996) can be used to find", "cite_spans": [ { "start": 142, "end": 156, "text": "Goodman (1996)", "ref_id": "BIBREF3" } ], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "arg max \u03c4 \u2208T (x) (a,i,j)\u2208\u03c4 \u00b5(a, i, j)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "This is the parsing algorithm used by Petrov et al. (2006) , for example. In addition, we can calculate the probability for an input sentence,", "cite_spans": [ { "start": 38, "end": 58, "text": "Petrov et al. (2006)", "ref_id": "BIBREF10" } ], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "p(x) = \u03c4 \u2208T (x) p(\u03c4 ), as p(x) = a\u2208I \u00b5(a, 1, N ).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "Variants of the inside-outside algorithm can be used for problems 1 and 2. This section introduces a novel form of these algorithms, using tensors. This is the first step in deriving the spectral estimation method.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "The algorithms are shown in figures 2 and 3. Each algorithm takes the following inputs: 1\u00d7m) for each rule a \u2192 x.", "cite_spans": [ { "start": 88, "end": 92, "text": "1\u00d7m)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "1. A tensor C a\u2192b c \u2208 R (m\u00d7m\u00d7m) for each rule a \u2192 b c. 2. A vector c \u221e a\u2192x \u2208 R (", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "3. A vector c 1 a \u2208 R (m\u00d71) for each a \u2208 I. The following theorem gives conditions under which the algorithms are correct:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "Theorem 1 Assume that we have an L-PCFG with parameters Q a\u2192x , Q a\u2192b c , T a\u2192b c , S a\u2192b c , \u03c0 a , and that there exist matrices G a \u2208 R (m\u00d7m) for all a \u2208 N such that each G a is invertible, and such that: Proof: See section 9.1.", "cite_spans": [ { "start": 138, "end": 143, "text": "(m\u00d7m)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "1. For all rules a \u2192 b c, C a\u2192b c (y) = G c T a\u2192b c diag(yG b S a\u2192b c )Q a\u2192b c (G a ) \u22121 2. For all rules a \u2192 x, c \u221e a\u2192x = 1 \u22a4 Q a\u2192x (G a ) \u22121 3. For all a \u2208 I, c 1 a = G a \u03c0", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tensor Form of the Inside-Outside Algorithm", "sec_num": "5" }, { "text": "A crucial result is that it is possible to directly estimate parameters C a\u2192b c , c \u221e a\u2192x and c 1 a that satisfy the conditions in theorem 1, from a training sample consisting of s-trees (i.e., trees where hidden variables are unobserved). We first describe random variables underlying the approach, then describe observable representations based on these random variables.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Estimating the Tensor Model", "sec_num": "6" }, { "text": "Each s-tree with N rules r 1 . . . r N has N nodes. We will use the s-tree in figure 1 as a running example.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Random Variables Underlying the Approach", "sec_num": "6.1" }, { "text": "Each node has an associated rule: for example, node 2 in the tree in figure 1 has the rule NP \u2192 D N. If the rule at a node is of the form a \u2192 b c, then there are left and right inside trees below the left child and right child of the rule. For example, for node 2 we have a left inside tree rooted at node 3, and a right inside tree rooted at node 4 (in this case the left and right inside trees both contain only a single rule production, of the form a \u2192 x; however in the general case they might be arbitrary subtrees).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Random Variables Underlying the Approach", "sec_num": "6.1" }, { "text": "In addition, each node has an outside tree. For node 2, the outside tree is (m\u00d71) for all a \u2208 I.", "cite_spans": [ { "start": 76, "end": 81, "text": "(m\u00d71)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Random Variables Underlying the Approach", "sec_num": "6.1" }, { "text": "S NP VP V saw P him Inputs: Sentence x1 . . . xN , L-PCFG (N , I, P, m, n), param- eters C a\u2192b c \u2208 R (m\u00d7m\u00d7m) for all a \u2192 b c \u2208 R, c \u221e a\u2192x \u2208 R (1\u00d7m) for all a \u2208 P, x \u2208 [n], c 1 a \u2208 R", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Random Variables Underlying the Approach", "sec_num": "6.1" }, { "text": "Data structures:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Random Variables Underlying the Approach", "sec_num": "6.1" }, { "text": "\u2022 Each \u03b1 a,i,j \u2208 R 1\u00d7m for a \u2208 N , 1 \u2264 i \u2264 j \u2264 N is a row vector of inside terms. \u2022 Each \u03b2 a,i,j \u2208 R m\u00d71 for a \u2208 N , 1 \u2264 i \u2264 j \u2264 N is a", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Random Variables Underlying the Approach", "sec_num": "6.1" }, { "text": "column vector of outside terms.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Random Variables Underlying the Approach", "sec_num": "6.1" }, { "text": "\u2022 Each \u00b5(a, i, j) \u2208 R for a \u2208 N , 1 \u2264 i \u2264 j \u2264 N is a marginal probability.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Random Variables Underlying the Approach", "sec_num": "6.1" }, { "text": "(Inside base case) \u2200a \u2208 P, i Figure 3 : The tensor form of the inside-outside algorithm, for calculation of marginal terms \u00b5(a, i, j).", "cite_spans": [], "ref_spans": [ { "start": 29, "end": 37, "text": "Figure 3", "ref_id": null } ], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2208 [N ], \u03b1 a,i,i = c \u221e a\u2192x i (Inside recursion) \u2200a \u2208 I, 1 \u2264 i < j \u2264 N, \u03b1 a,i,j = j\u22121 k=i a\u2192b c \u03b1 c,k+1,j C a\u2192b c (\u03b1 b,i,k ) (Outside base case) \u2200a \u2208 I, \u03b2 a,1,n = c 1 a (Outside recursion) \u2200a \u2208 N , 1 \u2264 i \u2264 j \u2264 N, \u03b2 a,i,j = i\u22121 k=1 b\u2192c a C b\u2192c a (\u03b1 c,k,i\u22121 )\u03b2 b,k,j + N k=j+1 b\u2192a c C b\u2192a c * (\u03b1 c,j+1,k )\u03b2 b,i,k (Marginals) \u2200a \u2208 N , 1 \u2264 i \u2264 j \u2264 N, \u00b5(a, i, j) = \u03b1 a,i,j \u03b2 a,i,j = h\u2208[m] \u03b1 a,i,j h \u03b2 a,i,j h", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "The outside tree contains everything in the s-tree r 1 . . . r N , excluding the subtree below node i.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "Our random variables are defined as follows. First, we select a random internal node, from a random tree, as follows:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022 Sample an s-tree r 1 . . . r N from the PMF p(r 1 . . . r N ). Choose a node i uniformly at random from [N ].", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "If the rule r i for the node i is of the form a \u2192 b c, we define random variables as follows:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022 R 1 is equal to the rule r i (e.g., NP \u2192 D N).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022 T 1 is the inside tree rooted at node i. T 2 is the inside tree rooted at the left child of node i, and T 3 is the inside tree rooted at the right child of node i.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022 H 1 , H 2 , H 3 are the hidden variables associated with node i, the left child of node i, and the right child of node i respectively.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022 A 1 , A 2 , A 3 are the labels for node i, the left child of node i, and the right child of node i respectively. (E.g., A 1 = NP, A 2 = D, A 3 = N.)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022 O is the outside tree at node i.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022 B is equal to 1 if node i is at the root of the tree (i.e., i = 1), 0 otherwise.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "If the rule r i for the selected node i is of the form a \u2192 x, we have random variables R 1 , T 1 , H 1 , A 1 , O, B as defined above, but H 2 , H 3 , T 2 , T 3 , A 2 , and A 3 are not defined.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "We assume a function \u03c8 that maps outside trees o to feature vectors \u03c8(o) \u2208 R d \u2032 . For example, the feature vector might track the rule directly above the node in question, the word following the node in question, and so on. We also assume a function \u03c6 that maps inside trees t to feature vectors \u03c6(t) \u2208 R d . As one example, the function \u03c6 might be an indicator function tracking the rule production at the root of the inside tree. Later we give formal criteria for what makes good definitions of \u03c8(o) of \u03c6(t). One requirement is that d \u2032 \u2265 m and d \u2265 m.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "In tandem with these definitions, we assume projection matices U a \u2208 R (d\u00d7m) and V a \u2208 R (d \u2032 \u00d7m) for all a \u2208 N . We then define additional random variables Y 1 , Y 2 , Y 3 , Z as", "cite_spans": [ { "start": 71, "end": 76, "text": "(d\u00d7m)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "Y 1 = (U a 1 ) \u22a4 \u03c6(T 1 ) Z = (V a 1 ) \u22a4 \u03c8(O) Y 2 = (U a 2 ) \u22a4 \u03c6(T 2 ) Y 3 = (U a 3 ) \u22a4 \u03c6(T 3 )", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "where a i is the value of the random variable A i . Note that Y 1 , Y 2 , Y 3 , Z are all in R m .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "Given the definitions in the previous section, our representation is based on the following matrix, tensor and vector quantities, defined for all a \u2208 N , for all rules of the form a \u2192 b c, and for all rules of the form a \u2192 x respectively:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "\u03a3 a = E[Y 1 Z \u22a4 |A 1 = a] D a\u2192b c = E [[R 1 = a \u2192 b c]]Y 3 Z \u22a4 Y \u22a4 2 |A 1 = a d \u221e a\u2192x = E [[R 1 = a \u2192 x]]Z \u22a4 |A 1 = a", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "Assuming access to functions \u03c6 and \u03c8, and projection matrices U a and V a , these quantities can be estimated directly from training data consisting of a set of s-trees (see section 7).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "Our observable representation then consists of:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "C a\u2192b c (y) = D a\u2192b c (y)(\u03a3 a ) \u22121 (2) c \u221e a\u2192x = d \u221e a\u2192x (\u03a3 a ) \u22121 (3) c 1 a = E [[[A 1 = a]]Y 1 |B = 1] (4)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "We next introduce conditions under which these quantities satisfy the conditions in theorem 1.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "The following definition will be important:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "Definition 2 For all a \u2208 N , we define the matrices I a \u2208 R (d\u00d7m) and J a \u2208", "cite_spans": [ { "start": 60, "end": 65, "text": "(d\u00d7m)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "R (d \u2032 \u00d7m) as [I a ] i,h = E[\u03c6 i (T 1 ) | H 1 = h, A 1 = a] [J a ] i,h = E[\u03c8 i (O) | H 1 = h, A 1 = a]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "In addition, for any a \u2208 N , we use \u03b3 a \u2208 R m to denote the vector with \u03b3 a h = P (H 1 = h|A 1 = a). The correctness of the representation will rely on the following conditions being satisfied (these are parallel to conditions 1 and 2 in Hsu et al. (2009) ):", "cite_spans": [ { "start": 238, "end": 255, "text": "Hsu et al. (2009)", "ref_id": "BIBREF4" } ], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "Condition 1 \u2200a \u2208 N , the matrices I a and J a are of full rank (i.e., they have rank m). For all a \u2208 N , for all h \u2208 [m], \u03b3 a h > 0. Condition 2 \u2200a \u2208 N , the matrices U a \u2208 R (d\u00d7m) and V a \u2208 R (d \u2032 \u00d7m) are such that the matrices G a = (U a ) \u22a4 I a and K a = (V a ) \u22a4 J a are invertible.", "cite_spans": [ { "start": 175, "end": 180, "text": "(d\u00d7m)", "ref_id": null }, { "start": 193, "end": 201, "text": "(d \u2032 \u00d7m)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "The following lemma justifies the use of an SVD calculation as one method for finding values for U a and V a that satisfy condition 2:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "Lemma 1 Assume that condition 1 holds, and for all a \u2208 N define", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "\u2126 a = E[\u03c6(T 1 ) (\u03c8(O)) \u22a4 |A 1 = a]", "eq_num": "(5)" } ], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "Then if U a is a matrix of the m left singular vectors of \u2126 a corresponding to non-zero singular values, and V a is a matrix of the m right singular vectors of \u2126 a corresponding to non-zero singular values, then condition 2 is satisfied.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "Proof sketch: It can be shown that \u2126 a = I a diag(\u03b3 a )(J a ) \u22a4 . The remainder is similar to the proof of lemma 2 in Hsu et al. (2009) .", "cite_spans": [ { "start": 118, "end": 135, "text": "Hsu et al. (2009)", "ref_id": "BIBREF4" } ], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "The matrices \u2126 a can be estimated directly from a training set consisting of s-trees, assuming that we have access to the functions \u03c6 and \u03c8.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "We can now state the following theorem:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "Theorem 2 Assume conditions 1 and 2 are satisfied. For all a \u2208 N , define G a = (U a ) \u22a4 I a . Then under the definitions in Eqs. 2-4:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "1. For all rules a \u2192 b c, C a\u2192b c (y) = G c T a\u2192b c diag(yG b S a\u2192b c )Q a\u2192b c (G a ) \u22121 2. For all rules a \u2192 x, c \u221e a\u2192x = 1 \u22a4 Q a\u2192x (G a ) \u22121 .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "3. For all a \u2208 N , c 1 a = G a \u03c0 a Proof: The following identities hold (see section 9.2):", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "D a\u2192b c (y) = (6) G c T a\u2192b c diag(yG b S a\u2192b c )Q a\u2192b c diag(\u03b3 a )(K a ) \u22a4 d \u221e a\u2192x = 1 \u22a4 Q a\u2192x diag(\u03b3 a )(K a ) \u22a4 (7) \u03a3 a = G a diag(\u03b3 a )(K a ) \u22a4 (8) c 1 a = G a \u03c0 a", "eq_num": "(9)" } ], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "Under conditions 1 and 2, \u03a3 a is invertible, and", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "(\u03a3 a ) \u22121 = ((K a ) \u22a4 ) \u22121 (diag(\u03b3 a )) \u22121 (G a ) \u22121 .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "The identities in the theorem follow immediately. Figure 4 shows an algorithm that derives estimates of the quantities in Eqs 2, 3, and 4. As input, the algorithm takes a sequence of tuples (r (i,1) , t (i,1) , t (i,2) , t (i,3) ,", "cite_spans": [ { "start": 213, "end": 218, "text": "(i,2)", "ref_id": null }, { "start": 223, "end": 228, "text": "(i,3)", "ref_id": null } ], "ref_spans": [ { "start": 50, "end": 58, "text": "Figure 4", "ref_id": null } ], "eq_spans": [], "section": "Observable Representations", "sec_num": "6.2" }, { "text": "o (i) , b (i) ) for i \u2208 [M ].", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "These tuples can be derived from a training set consisting of s-trees \u03c4 1 . . . \u03c4 M as follows:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "\u2022 \u2200i \u2208 [M ]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": ", choose a single node j i uniformly at random from the nodes in \u03c4 i . Define r (i,1) to be the rule at node j i . t (i,1) is the inside tree rooted at node j i . If r (i,1) is of the form a \u2192 b c, then t (i,2) is the inside tree under the left child of node j i , and t (i,3) is the inside tree under the right child of node j i . If r (i,1) is of the form a \u2192 x, then t (i,2) = t (i,3) = NULL. o (i) is the outside tree at node j i . b (i) is 1 if node j i is at the root of the tree, 0 otherwise.", "cite_spans": [ { "start": 80, "end": 85, "text": "(i,1)", "ref_id": null }, { "start": 117, "end": 122, "text": "(i,1)", "ref_id": null }, { "start": 168, "end": 173, "text": "(i,1)", "ref_id": null }, { "start": 205, "end": 210, "text": "(i,2)", "ref_id": null }, { "start": 271, "end": 276, "text": "(i,3)", "ref_id": null }, { "start": 337, "end": 342, "text": "(i,1)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "Under this process, assuming that the s-trees \u03c4 1 . . . \u03c4 M are i.i.d. draws from the distribution p(\u03c4 ) over s-trees under an L-PCFG, the tuples (r (i,1) , t (i,1) , t (i,2) , t (i,3) , o (i) , b (i) ) are i.i.d. draws from the joint distribution over the random variables R 1 , T 1 , T 2 , T 3 , O, B defined in the previous section.", "cite_spans": [ { "start": 169, "end": 174, "text": "(i,2)", "ref_id": null }, { "start": 179, "end": 184, "text": "(i,3)", "ref_id": null }, { "start": 189, "end": 192, "text": "(i)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "The algorithm first computes estimates of the projection matrices U a and V a : following lemma 1, this is done by first deriving estimates of \u2126 a , and then taking SVDs of each \u2126 a . The matrices are then used to project inside and outside trees t (i,1) , t (i,2) , t (i,3) , o (i) down to m-dimensional vectors y (i,1) , y (i,2) , y (i,3) , z (i) ; these vectors are used to derive the estimates of C a\u2192b c , c \u221e a\u2192x , and c 1 a . We now state a PAC-style theorem for the learning algorithm. First, for a given L-PCFG, we need a couple of definitions:", "cite_spans": [ { "start": 259, "end": 264, "text": "(i,2)", "ref_id": null }, { "start": 269, "end": 274, "text": "(i,3)", "ref_id": null }, { "start": 325, "end": 330, "text": "(i,2)", "ref_id": null }, { "start": 335, "end": 340, "text": "(i,3)", "ref_id": null }, { "start": 345, "end": 348, "text": "(i)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "\u2022 \u039b is the minimum absolute value of any element of the vectors/matrices/tensors c 1 a , d \u221e a\u2192x , D a\u2192b c , (\u03a3 a ) \u22121 . (Note that \u039b is a function of the projection matrices U a and V a as well as the underlying L-PCFG.)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "\u2022 For each a \u2208 N , \u03c3 a is the value of the m'th largest singular value of \u2126 a . Define \u03c3 = min a \u03c3 a .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "We then have the following theorem:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "Theorem 3 Assume that the inputs to the algorithm in figure 4 are i.i.d. draws from the joint distribution over the random variables figure 3 with inputs\u0109 1 a ,\u0109 \u221e a\u2192x ,\u0108 a\u2192b c derived from the algorithm in figure 4. Define R to be the total number of rules in the grammar of the form a \u2192 b c or a \u2192 x. Define M a to be the number of training examples in the input to the algorithm in figure 4 where r i,1 has non-terminal a on its lefthand-side. Under these assumptions, if for all a", "cite_spans": [], "ref_spans": [ { "start": 133, "end": 156, "text": "figure 3 with inputs\u0109 1", "ref_id": "FIGREF1" } ], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "R 1 , T 1 , T 2 , T 3 , O, B,", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "M a \u2265 128m 2 2N+1 \u221a 1 + \u01eb \u2212 1 2 \u039b 2 \u03c3 4 log 2mR \u03b4 Then 1 \u2212 \u01eb \u2264 p(r 1 . . . r N ) p(r 1 . . . r N ) \u2264 1 + \u01eb", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "A similar theorem (omitted for space) states that 1 \u2212 \u01eb \u2264 \u03bc (a,i,j) \u00b5(a,i,j) \u2264 1 + \u01eb for the marginals. The condition that\u00db a andV a are derived from \u2126 a , as opposed to the sample estimate\u03a9 a , follows Foster et al. (2012) . As these authors note, similar techniques to those of Hsu et al. (2009) should be applicable in deriving results for the case where\u03a9 a is used in place of \u2126 a .", "cite_spans": [ { "start": 60, "end": 67, "text": "(a,i,j)", "ref_id": null }, { "start": 203, "end": 223, "text": "Foster et al. (2012)", "ref_id": "BIBREF2" }, { "start": 280, "end": 297, "text": "Hsu et al. (2009)", "ref_id": "BIBREF4" } ], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "Proof sketch: The proof is similar to that of Foster et al. (2012) . The basic idea is to first show that under the assumptions of the theorem, the estimate\u015d c 1 a ,d \u221e a\u2192x ,D a\u2192b c ,\u03a3 a are all close to the underlying values being estimated. The second step is to show that this ensures thatp", "cite_spans": [ { "start": 46, "end": 66, "text": "Foster et al. (2012)", "ref_id": "BIBREF2" } ], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "(r 1 ...r N \u2032 ) p(r 1 ...r N \u2032 )", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "is close to 1. The method described of selecting a single tuple (r (i,1) , t (i,1) , t (i,2) , t (i,3) , o (i) , b (i) ) for each s-tree ensures that the samples are i.i.d., and simplifies the analysis underlying theorem 3. In practice, an implementation should most likely use all nodes in all trees in training data; by Rao-Blackwellization we know such an algorithm would be better than the one presented, but the analysis of how much better would be challenging. It would almost certainly lead to a faster rate of convergence ofp to p.", "cite_spans": [ { "start": 67, "end": 72, "text": "(i,1)", "ref_id": null }, { "start": 77, "end": 82, "text": "(i,1)", "ref_id": null }, { "start": 87, "end": 92, "text": "(i,2)", "ref_id": null }, { "start": 97, "end": 102, "text": "(i,3)", "ref_id": null }, { "start": 107, "end": 110, "text": "(i)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Deriving Empirical Estimates", "sec_num": "7" }, { "text": "There are several potential applications of the method. The most obvious is parsing with L-PCFGs. 1 The approach should be applicable in other cases where EM has traditionally been used, for example in semi-supervised learning. Latent-variable HMMs for sequence labeling can be derived as special case of our approach, by converting tagged sequences to right-branching skeletal trees.", "cite_spans": [ { "start": 98, "end": 99, "text": "1", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Discussion", "sec_num": "8" }, { "text": "The sample complexity of the method depends on the minimum singular values of \u2126 a ; these singular values are a measure of how well correlated \u03c8 and \u03c6 are with the unobserved hidden variable H 1 . Experimental work is required to find a good choice of values for \u03c8 and \u03c6 for parsing.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Discussion", "sec_num": "8" }, { "text": "This section gives proofs of theorems 1 and 2. Due to space limitations we cannot give full proofs; instead we provide proofs of some key lemmas. A long version of this paper will give the full proofs.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Proofs", "sec_num": "9" }, { "text": "First, the following lemma leads directly to the correctness of the algorithm in figure 2:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Proof of Theorem 1", "sec_num": "9.1" }, { "text": "1 Parameters can be estimated using the algorithm in figure 4 ; for a test sentence x1 . . . xN we can first use the algorithm in figure 3 to calculate marginals \u00b5(a, i, j), then use the algorithm of Goodman (1996) to find arg max \u03c4 \u2208T (x) (a,i,j)\u2208\u03c4 \u00b5(a, i, j).", "cite_spans": [ { "start": 200, "end": 214, "text": "Goodman (1996)", "ref_id": "BIBREF3" } ], "ref_spans": [ { "start": 53, "end": 61, "text": "figure 4", "ref_id": null } ], "eq_spans": [], "section": "Proof of Theorem 1", "sec_num": "9.1" }, { "text": "Inputs: Training examples (r (i,1) , t (i,1) , t (i,2) , t (i,3) , o (i) , b (i) ) for i \u2208 {1 . . . M }, where r (i,1) is a context free rule; t (i,1) , t (i,2) and t (i,3) are inside trees; o (i) is an outside tree; and b (i) = 1 if the rule is at the root of tree, 0 otherwise. A function \u03c6 that maps inside trees t to feature-vectors \u03c6(t) \u2208 R d . A function \u03c8 that maps outside trees o to feature-vectors \u03c8(o) \u2208 R d \u2032 .", "cite_spans": [ { "start": 29, "end": 34, "text": "(i,1)", "ref_id": null }, { "start": 39, "end": 44, "text": "(i,1)", "ref_id": null }, { "start": 49, "end": 54, "text": "(i,2)", "ref_id": null }, { "start": 59, "end": 64, "text": "(i,3)", "ref_id": null }, { "start": 69, "end": 72, "text": "(i)", "ref_id": null }, { "start": 155, "end": 160, "text": "(i,2)", "ref_id": null }, { "start": 167, "end": 172, "text": "(i,3)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Proof of Theorem 1", "sec_num": "9.1" }, { "text": "Define ai to be the non-terminal on the left-hand side of rule r (i,1) . If r (i,1) is of the form a \u2192 b c, define bi to be the nonterminal for the left-child of r (i,1) , and ci to be the non-terminal for the right-child. (Step 0: Singular Value Decompositions)", "cite_spans": [ { "start": 65, "end": 70, "text": "(i,1)", "ref_id": null }, { "start": 78, "end": 83, "text": "(i,1)", "ref_id": null }, { "start": 164, "end": 169, "text": "(i,1)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022 Use the algorithm in figure 5 to calculate matrices\u00db a \u2208", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "R (d\u00d7m) andV a \u2208 R (d \u2032 \u00d7m) for each a \u2208 N . (Step 1: Projection) \u2022 For all i \u2208 [M ], compute y (i,1) = (\u00db a i ) \u22a4 \u03c6(t (i,1) ).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022 For all i \u2208 [M ] such that r (i,1) is of the form a \u2192 b c, compute y (i,2) = (\u00db b i ) \u22a4 \u03c6(t (i,2) ) and y (i,3) = (\u00db c i ) \u22a4 \u03c6(t (i,3) ).", "cite_spans": [ { "start": 31, "end": 36, "text": "(i,1)", "ref_id": null }, { "start": 94, "end": 99, "text": "(i,2)", "ref_id": null }, { "start": 131, "end": 136, "text": "(i,3)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "For all i \u2208 [M ], compute z (i) = (V a i ) \u22a4 \u03c8(o (i) ).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "(", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "Step 2: Calculate Correlations)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022 For each a \u2208 N , define", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u03b4a = 1/ M i=1 [[ai = a]] \u2022 For each rule a \u2192 b c, computeD a\u2192b c = \u03b4a \u00d7 M i=1 [[r (i,1) = a \u2192 b c]]y (i,3) (z (i) ) \u22a4 (y (i,2) ) \u22a4 \u2022 For each rule a \u2192 x, computed \u221e a\u2192x = \u03b4a \u00d7 M i=1 [[r (i,1) = a \u2192 x]](z (i) ) \u22a4 \u2022 For each a \u2208 N , compute\u03a3 a = \u03b4a \u00d7 M i=1 [[ai = a]]y (i,1) (z (i) ) \u22a4 (Step 3: Compute Final Parameters) \u2022 For all a \u2192 b c,\u0108 a\u2192b c (y) =D a\u2192b c (y)(\u03a3 a ) \u22121 \u2022 For all a \u2192 x,\u0109 \u221e a\u2192x =d \u221e a\u2192x (\u03a3 a ) \u22121 \u2022 For all a \u2208 I,\u0109 1 a = M i=1 [[a i =a and b (i) =1]]y (i,1) M i=1 [[b (i) =1]]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "Figure 4: The spectral learning algorithm.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "Inputs: Identical to algorithm in figure 4. Algorithm:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022 For each a \u2208 N , compute\u03a9 a \u2208 R (d \u2032 \u00d7d) a\u015d", "cite_spans": [], "ref_spans": [ { "start": 39, "end": 42, "text": "\u00d7d)", "ref_id": null } ], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2126 a = M i=1 [[ai = a]]\u03c6(t (i,1) )(\u03c8(o (i) )) \u22a4 M i=1 [[ai = a]]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "and calculate a singular value decomposition of\u03a9 a .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "\u2022 For each a \u2208 N , define\u00db a \u2208 R m\u00d7d to be a matrix of the left singular vectors of\u03a9 a corresponding to the m largest singular values. DefineV a \u2208 R m\u00d7d \u2032 to be a matrix of the right singular vectors of\u03a9 a corresponding to the m largest singular values. Lemma 3 Assume that conditions 1-3 of theorem 1 are satisfied, and that the input to the algorithm in figure 3 is a sentence x 1 . . . x N . For any a \u2208 N , for any 1 \u2264 i \u2264 j \u2264 N , define\u1fb1 a,i,j \u2208 R (1\u00d7m) to have components\u1fb1 a,i,j h = p(x i . . . x j |h, a) for h \u2208 [m]. In addition, define\u03b2 a,i,j \u2208 R (m\u00d71) to have components\u03b2 a,i,j h = p(x 1 . . . x i\u22121 , a(h), x j+1 . . . x N ) for h \u2208 [m] . Then for all i \u2208 [N ], \u03b1 a,i,j =\u1fb1 a,i,j (G a ) \u22121 and \u03b2 a,i,j = G a\u03b2a,i,j . It follows that for all (a, i, j), \u00b5(a, i, j) =\u1fb1 a,i,j (G a ) \u22121 G a\u03b2a,i,j =\u1fb1 a,i,j\u03b2a,i,j Thus the vectors \u03b1 a,i,j and \u03b2 a,i,j are linearly related to the vectors\u1fb1 a,i,j and\u03b2 a,i,j , which are the inside and outside terms calculated by the conventional form of the inside-outside algorithm.", "cite_spans": [ { "start": 644, "end": 647, "text": "[m]", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "The proof is by induction, and is similar to the proof of lemma 2; for reasons of space it is omitted.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm:", "sec_num": null }, { "text": "We now prove the identity in Eq. 6, used in the proof of theorem 2. For reasons of space, we do not give the proofs of identities 7-9: the proofs are similar.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Proof of the Identity in Eq. 6", "sec_num": "9.2" }, { "text": "The following identities can be verified: where E a\u2192b c = G c T a\u2192b c , F a\u2192b c = G b S a\u2192b c . Y 3 , Z and Y 2 are independent when conditioned on H 1 , R 1 (this follows from the independence assumptions in the L-PCFG), hence", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Proof of the Identity in Eq. 6", "sec_num": "9.2" }, { "text": "P (R 1 =", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Proof of the Identity in Eq. 6", "sec_num": "9.2" }, { "text": "E [[[R 1 = a \u2192 b c]]Y 3,j Z k Y 2,l | H 1 = h, A 1 = a] = q(a \u2192 b c|h, a)E a\u2192b c j,h K a k,h F a\u2192b c l,h", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Proof of the Identity in Eq. 6", "sec_num": "9.2" }, { "text": "Hence (recall that \u03b3 a h = P (H 1 = h|A 1 = a)), ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Proof of the Identity in Eq. 6", "sec_num": "9.2" }, { "text": "from which Eq. 6 follows.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Proof of the Identity in Eq. 6", "sec_num": "9.2" } ], "back_matter": [ { "text": "Acknowledgements: Columbia University gratefully acknowledges the support of the Defense Advanced Research Projects Agency (DARPA) Machine Reading Program under Air Force Research Laboratory (AFRL) prime contract no. FA8750-09-C-0181. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the view of DARPA, AFRL, or the US government. Shay Cohen was supported by the National Science Foundation under Grant #1136996 to the Computing Research Association for the CIFellows Project. Dean Foster was supported by National Science Foundation grant 1106743.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "acknowledgement", "sec_num": null }, { "text": "Lemma 2 Assume that conditions 1-3 of theorem 1 are satisfied, and that the input to the algorithm in figure 2 is an s-tree r 1 . . . r N . Define a i for i \u2208 [N ] to be the non-terminal on the left-hand-side of rule r i , and t i for i \u2208 [N ] to be the s-tree with rule r i at its root. Finally, for all i \u2208 [N ], define the row vector b i \u2208 R (1\u00d7m) to have componentsThis lemma shows a direct link between the vectors f i calculated in the algorithm, and the terms b i h , which are terms calculated by the conventional inside algorithm: each f i is a linear transformation (through G a i ) of the corresponding vector b i . Proof: The proof is by induction.First consider the base case. For any leaf-i.e., for any i such that a i \u2208 P-we have b i h = q(r i |h, a i ), and it is easily verified thatThe inductive case is as follows. For all i \u2208 [N ] such that a i \u2208 I, by the definition in the algorithm,Assuming by induction that f \u03b3 = b \u03b3 (G (a\u03b3 ) ) \u22121 and f \u03b2 = b \u03b2 (G (a \u03b2 ) ) \u22121 , this simplifies towhere \u03ba r = b \u03b3 T r i , and, hence \u03ba r diag(\u03ba l )Q r i = b i and the inductive case follows immediately from Eq. 10.Next, we give a similar lemma, which implies the correctness of the algorithm in figure 3:", "cite_spans": [ { "start": 345, "end": 350, "text": "(1\u00d7m)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "annex", "sec_num": null } ], "bib_entries": { "BIBREF0": { "ref_id": "b0", "title": "A spectral learning algorithm for finite state transducers", "authors": [ { "first": "B", "middle": [], "last": "Balle", "suffix": "" }, { "first": "A", "middle": [], "last": "Quattoni", "suffix": "" }, { "first": "X", "middle": [], "last": "Carreras", "suffix": "" } ], "year": 2011, "venue": "Proceedings of ECML", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "B. 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For each a \u2192 b c \u2208 R, and h, h \u2032 \u2208 [m], we have parameters s(h \u2032 |h, a \u2192 b c) and t(h \u2032 |h, a \u2192 b c).", "type_str": "figure" }, "FIGREF1": { "uris": null, "num": null, "text": "An s-tree, and its sequence of rules. (For convenience we have numbered the nodes in the tree.) the hidden variable for the left-hand-side of rule r i . Each h i can take any value in [m].", "type_str": "figure" }, "FIGREF2": { "uris": null, "num": null, "text": "The tensor form for calculation of p(r 1 . . . r N ).", "type_str": "figure" }, "FIGREF3": { "uris": null, "num": null, "text": "a Then: 1) The algorithm in figure 2 correctly computes p(r 1 . . . r N ) under the L-PCFG. 2) The algorithm in figure 3 correctly computes the marginals \u00b5(a, i, j) under the L-PCFG.", "type_str": "figure" }, "FIGREF4": { "uris": null, "num": null, "text": "Singular value decompositions.", "type_str": "figure" }, "FIGREF6": { "uris": null, "num": null, "text": "a \u2192 b c|H 1 = h, A 1 = a) = q(a \u2192 b c|h, a) E [Y 3,j |H 1 = h, R 1 = a \u2192 b c] = E a\u2192b c j,h E [Z k |H 1 = h, R 1 = a \u2192 b c] = K a k,h E [Y 2,l |H 1 = h, R 1 = a \u2192 b c] = F a\u2192b c l,h", "type_str": "figure" }, "FIGREF7": { "uris": null, "num": null, "text": "D a\u2192b c j,k,l = E [[[R 1 = a \u2192 b c]]Y 3,j Z k Y 2,l | A 1 = a] = h \u03b3 a h E [[[R 1 = a \u2192 b c]]Y 3,j Z k Y 2,l | H 1 = h, A 1 = a] = h \u03b3 a h q(a \u2192 b c|h, a)E a\u2192b c j,h K a k,h F a\u2192b c l,h", "type_str": "figure" }, "TABREF0": { "text": "For any s-tree r 1 . . . r N definep(r 1 . . . r N ) to be the value calculated by the algorithm in", "type_str": "table", "html": null, "content": "
un-
der an L-PCFG with distribution p(r 1 . . . r N ) over
s-trees. Define m to be the number of latent states
in the L-PCFG. Assume that the algorithm in fig-
ure 4 has projection matrices\u00db a andV a derived as
left and right singular vectors of \u2126 a , as defined in
Eq. 5. Assume that the L-PCFG, together with\u00db a
andV a , has coefficients \u039b > 0 and \u03c3 > 0. In addi-
tion, assume that all elements in c 1 a , d \u221e a\u2192x , D a\u2192b c , and \u03a3 a are in [\u22121, +1].
", "num": null } } } }