acl acl2013 acl2013-332 knowledge-graph by maker-knowledge-mining

332 acl-2013-Subtree Extractive Summarization via Submodular Maximization


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Author: Hajime Morita ; Ryohei Sasano ; Hiroya Takamura ; Manabu Okumura

Abstract: This study proposes a text summarization model that simultaneously performs sentence extraction and compression. We translate the text summarization task into a problem of extracting a set of dependency subtrees in the document cluster. We also encode obligatory case constraints as must-link dependency constraints in order to guarantee the readability of the generated summary. In order to handle the subtree extraction problem, we investigate a new class of submodular maximization problem, and a new algorithm that has the approximation ratio 21 (1 − e−1). Our experiments with the NTC(1IR − −A eCLIA test collections show that our approach outperforms a state-of-the-art algorithm.

Reference: text


Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 We translate the text summarization task into a problem of extracting a set of dependency subtrees in the document cluster. [sent-9, score-0.44]

2 We also encode obligatory case constraints as must-link dependency constraints in order to guarantee the readability of the generated summary. [sent-10, score-0.267]

3 In order to handle the subtree extraction problem, we investigate a new class of submodular maximization problem, and a new algorithm that has the approximation ratio 21 (1 − e−1). [sent-11, score-1.004]

4 1 Introduction Text summarization is often addressed as a task of simultaneously performing sentence extraction and sentence compression (Berg-Kirkpatrick et al. [sent-13, score-0.445]

5 Joint models of sentence extraction and compression have a great benefit in that they have a large degree of freedom as far as controlling redundancy goes. [sent-15, score-0.267]

6 Meanwhile, submodular maximization has recently been applied to the text summarization task, and the methods thereof have performed very well (Lin and Bilmes, 2010; Lin and Bilmes, 2011; Morita et al. [sent-29, score-0.732]

7 Formalizing summarization as a submodular maximization problem has an important benefit inthat the problem can be solved by using a greedy algorithm with a performance guarantee. [sent-31, score-0.824]

8 We therefore decided to formalize the task of simultaneously performing sentence extraction and compression as a submodular maximization problem. [sent-32, score-0.939]

9 The existing greedy algorithm for solving submodular maximization problems cannot work in the presence of such numerous constraints although monotone and nonmonotone submodular maximization with constraints other than budget constraints have been studied (Lee et al. [sent-36, score-1.655]

10 In this study, we avoid this difficulty by reducing the task to one of extracting dependency subtrees from sentences in the source documents. [sent-40, score-0.347]

11 The reduction replaces the difficulty of numerous linear constraints with another difficulty wherein two subtrees can share the same word to1023 Proce dingsS o f ita h,e B 5u1lgsta Arinan,u Aaulg Musete 4ti-n9g 2 o0f1 t3h. [sent-41, score-0.391]

12 Ac s2s0o1ci3a Atiosnso fcoirat Cio nm foprut Caotimonpaulta Lti nognuails Lti cnsg,u piasgteics 1023–1032, ken when they are selected from the same sentence, and as a result, the cost of the union of the two subtrees is not always the mere sum of their costs. [sent-43, score-0.642]

13 We can overcome this difficulty by tackling a new class of submodular maximization problem: a budgeted monotone nondecreasing submodular function maximization with a cost function, where the cost of an extraction unit varies depending on what other extraction units are selected. [sent-44, score-1.994]

14 We also propose a new greedy algorithm that solves this new class of maximization problem with a performance guarantee 12(1 e−1). [sent-46, score-0.38]

15 This property is suitable for summarization purposes, because the gain of adding a new sentence to a summary that already contains sufficient information should be small. [sent-55, score-0.34]

16 Therefore, many studies have formalized text summarization as a submodular maximization problem (Lin and Bilmes, 2010; Lin and Bilmes, 2011; Morita et ≤ + − ≤ − al. [sent-56, score-0.732]

17 To our knowledge, there is no study that addresses the joint task of simultaneously performing compression and extraction through an approximate submodular maximization with a performance guarantee. [sent-59, score-0.887]

18 (2009) proposed an algorithm that solves the submodular maximization problem under multiple linear constraints with a performance guarantee 1− e−1 in polynomial time. [sent-61, score-0.81]

19 (201 1) formulated a unified task of sentence extraction and sentence compression as an ILP. [sent-65, score-0.319]

20 1 Problem Definition Let V be the finite set of all valid subtrees in the source documents, where valid subtrees are defined to be the ones that can be regarded as grammatical sentences. [sent-68, score-0.9]

21 In this paper, we regard subtrees containing the root node of the sentence as valid. [sent-69, score-0.48]

22 Accordingly, V denotes a set of all rooted subtrees in all sentences. [sent-70, score-0.407]

23 A subtree contains a set of elements that are units in a dependency structure (e. [sent-71, score-0.27]

24 This means the cost of a subtree is the integer number of characters it contains. [sent-77, score-0.409]

25 V is partitioned into exclusive subsets B ofvalid subtrees, and each subset corresponds to the original sentence from which the valid subtrees derived. [sent-78, score-0.502]

26 However, the cost of a union of subtrees from different sentences is simply the sum of the costs of subtrees, while the cost of a union of subtrees from the same sentence is smaller than the sum of the costs. [sent-79, score-1.336]

27 (1) For example, if we∑ Badd a subtree t containing words {wa, wb, wc} to a summary that already covers {wwords {wa, wb, wd} fmromma ythe th same sentence, sth weo raddsdit {iownal cost of} }t f riso only c({wc}) eb en-cause wa a anddd wb are already tc oisv oernelyd1 . [sent-81, score-0.556]

28 The first requirement is that the union of valid subtrees is also a valid subtree. [sent-83, score-0.673]

29 The second requirement is that the union of subtrees and a single valid subtree have the same score and the same cost if they cover the same elements. [sent-84, score-1.011]

30 We will refer to the single valid subtree as the equivalent subtree of the union of subtrees. [sent-85, score-0.763]

31 These requirements enable us to represent sentence compression as the extraction of subtrees from a sentence. [sent-86, score-0.614]

32 This is because the requirements guarantee that the extracted subtrees represent a sentence. [sent-87, score-0.42]

33 U means the set of subtrees that are not extracted. [sent-96, score-0.347]

34 The algorithm iteratively adds to the current summary the element si that has the largest ratio of the objective function gain to the additional cost, unless adding it violates the budget constraint. [sent-97, score-0.507]

35 V implicitly constrains the model such that it can only select valid subtrees from a set of nodes and edges. [sent-102, score-0.45]

36 The performance guarantee oftheir algorithm is actually the same as ours, since Algorithm 1Modified greedy algorithm for budgeted submodular function maximization with a cost function . [sent-107, score-1.243]

37 3 Relation with Discrete Optimization We argue that our optimization problem can be regarded as an extraction of subtrees rooted at a given node from a directed graph, instead of from a tree. [sent-112, score-0.493]

38 However, while matroids are often used to represent constraints on submodular maximization problems (Conforti and Cornu ´ejols, 1984; Calinescu et al. [sent-116, score-0.714]

39 To our knowledge, this is the first study that gives a constant performance guarantee for the submodular maximization under greedoid (non-matroid) constraints. [sent-118, score-0.789]

40 1025 4 Joint Model of Extraction and Compression We will formalize the unified task of sentence compression and extraction as a budgeted monotone nondecreasing submodular function maximization with a cost function. [sent-128, score-1.401]

41 In this formalization, a valid subtree of a sentence represents a candidate of a compressed sentence. [sent-129, score-0.511]

42 We will refer to all valid subtrees of a given sentence as a valid set. [sent-130, score-0.605]

43 A valid set corresponds to all candidates of the compression of a sentence. [sent-131, score-0.277]

44 Since, from the requirements, the union of valid subtrees is also a valid subtree in the valid set, the model can extract one or more subtrees from one sentence, and generate a compressed sentence by merging those subtrees to generate an equivalent subtree. [sent-133, score-1.878]

45 Therefore, the joint model can extract an arbitrarily compressed sentence as a subtree without enumerating all candidates. [sent-134, score-0.443]

46 We can approximately solve the subtree extraction problem by using Algorithm 1. [sent-136, score-0.311]

47 On line 5 of the algorithm, the subtree extraction is performed as a local search that finds maximal density subtrees from the whole documents. [sent-137, score-0.872]

48 The maximal density subtree is a subtree that has the highest score per cost of subtree. [sent-138, score-0.888]

49 In this paper, we address the task of summarization of Japanese text by means of sentence compression and extraction. [sent-140, score-0.319]

50 In Japanese, syntactic subtrees that contain the root of the dependency tree of the original sentence often make grammatical sentences. [sent-141, score-0.466]

51 1 that a union of valid subtrees is a valid and equivalent tree is of- ten true for Japanese. [sent-143, score-0.704]

52 In this joint model, we generate a compressed sentence by extracting an arbitrary subtree from a dependency tree of a sentence. [sent-150, score-0.439]

53 The sentence generated by a subtree can be unnatural even though the subtree contains the root node of the sentence. [sent-152, score-0.673]

54 1 Objective Function We extract subtrees from sentences in order to solve the query-oriented summarization problem as a unified one consisting of sentence compression and extraction. [sent-162, score-0.666]

55 With such a similarity, sentence compression extracts nearly only the query terms and fails to contain important information. [sent-165, score-0.292]

56 The score function is supermodular as a score function of subtree extraction3, because the union of two subtrees can have extra word pairs that are not included in either subtree. [sent-170, score-0.941]

57 The behavior can be represented by a submodular objective function that reduces word scores depending on those already included in the summary. [sent-177, score-0.56]

58 Our positive reward for long sentences is represented as reward(S) = c(S) |S|, (3) where c(S) is the cost of summary S, and |S| is the − wnuhmerbeerc oSf) siesnttheenccoess tionf Ssu. [sent-181, score-0.355]

59 Let d be the damping rate, countS(w) be the number of sentences containing word w in summary S, words(S) be the set of words included in summary S, qsb(w) be the query relevance score of word w, and γ be a parameter that adjusts the rate of sentence compression. [sent-183, score-0.384]

60 It is also ad3The score is still submodular for the purpose of sentence extraction. [sent-186, score-0.508]

61 vantageous that the submodular maximization can deal with such objective functions. [sent-187, score-0.705]

62 Due to the nature of the objective function, we can use dynamic programming to effectively search for the subtree with the maximal density. [sent-189, score-0.41]

63 We will use a fast algorithm to find the maximal density subtree (MDS) of a given sentence for each cost in Algorithm 1. [sent-192, score-0.692]

64 That is, the gain function of adding a subtree to a summary can be represented as the sum of gains for words: g(t) = ∑{gainS(w) + freqt(w)c(w)γ}, ∑w∈t gainS(w) = qsb(w)dcountS(w) , where freqt(w) is the number of ws in subtree t, and gainS(w) is the gain of adding the word w to the summary S. [sent-195, score-1.069]

65 Our algorithm is based on dynamic programming, and it selects a subtree that maximizes the gain function per cost. [sent-196, score-0.472]

66 We extended this algorithm to work for submodular word gain functions that are not constant. [sent-199, score-0.556]

67 Let n be a node in the tree, a and b be child nodes of n, c(n) be the cost of n, mdsac be the MDS rooted at {mdscn(n), a and have cost c. [sent-204, score-0.383]

68 The valid subtrees rooted at n can be ob- tained by taking unions of n with one or both of t1 ∈ mdsa and t2 ∈ mdsb. [sent-209, score-0.551]

69 mdscn is the union that has∈ ∈th me largest gain over sthe union with the cost of c (by enumerating all the unions). [sent-210, score-0.569]

70 Next, let us consider the objective function that returns the sum of values of submodular word gain functions. [sent-212, score-0.706]

71 s∪u mbt dien-dicates a set of current maximal density subtrees among the combinations calculated before. [sent-225, score-0.524]

72 newt indicates a set of temporary maximal density subtrees for the combinations calculated from line 4 to 8. [sent-226, score-0.629]

73 subt[cost,ws] indicates a element of subt that has a cost cost and contains a set of words ws. [sent-227, score-0.436]

74 Line 1 sets subt to a set consisting of a subtree that indicates node n itself. [sent-229, score-0.438]

75 The algorithm calculates maximal density subtrees within combinations of the root node n and MDSs rooted at child nodes of n. [sent-230, score-0.753]

76 Line 3 iteratively adds MDSs rooted at a next child node to the combinations; the algorithm then calculates MDSs newt between subt and the MDSs of the child node. [sent-231, score-0.384]

77 The procedure from line 6 to 8 selects a subtree that has a larger gain from the temporary maximal subtree and the union of t and m. [sent-232, score-0.849]

78 When we treat word overlaps, we need to count Algorithm 2Algorithm for finding maximal density subtree for each cost: MDSs. [sent-236, score-0.447]

79 The right table enumerates the subtrees rooted at w2 in the left tree for all indices. [sent-238, score-0.438]

80 Lin and Bilmes (201 1) designed a monotone submodular function for query-oriented summarization. [sent-275, score-0.613]

81 They proposed a positive diversity reward function in order to define a monotone submodular objective function for generating a non-redundant summary. [sent-277, score-0.848]

82 The diversity reward gives a smaller gain for a biased summary, because it consists of gains based on three clusters and calculates a square root score with respect to each sentence. [sent-278, score-0.312]

83 The reward also contains a score for the similarity of a sentence to the query, for purposes of query-oriented summa- RecallLength# of nuggets Subtree extraction0. [sent-279, score-0.271]

84 In the coverage function min function limits the maximum gain α ∑i∈V wi,j, which is a small fraction α of the sim∑ilarity between a sentence j and the all source d∑ocuments. [sent-284, score-0.27]

85 7 Conclusions and Future Work We formalized a query-oriented summarization, which is a task in which one simultaneously performs sentence compression and extraction, as a new optimization problem: budgeted monotone nondecreasing submodular function maximization with a cost function. [sent-321, score-1.393]

86 2 % improvement over a state-of-the-art method using a submodular objective function. [sent-325, score-0.49]

87 Since our algorithm requires that the objective function is the sum of word score functions, our proposed method has a restriction that we cannot use an arbitrary monotone submodular function as the objective function for the summary. [sent-326, score-1.007]

88 S∗ is the optimal solution, cu(S) is the residual cost of − subtree u when S is already covered, and i∗ is the last step before the algorithm discards a subtree s ∈ S∗ or a part of the subtree s. [sent-331, score-1.003]

89 We can remove the subtree s0 from V without changing the approximate rate. [sent-333, score-0.27]

90 si is the i-th subtree obtained by line 5 of Algorithm 1. [sent-334, score-0.36]

91 Gi is the set obtained after adding subtree si to Gi−1 from the valid set Bi. [sent-335, score-0.426]

92 f(·) : 2V → R is a monotone submodular funcftio(·n). [sent-337, score-0.543]

93 We assume that there is an equivalent subtree with any union of subtrees in a valid set B: ∀t1 , t2, ∃te, te ≡ {t1, t2}. [sent-338, score-0.84]

94 Let B be a valid set, and union be a function that returns the union of subtrees. [sent-351, score-0.413]

95 Lemma 3 For a normalized monotone submodular f(·), for i= 1, . [sent-366, score-0.543]

96 Maximizing a monotone submodular function subject to a matroid constraint. [sent-383, score-0.659]

97 The weight-constrained maximum-density subtree problem and related problems in trees. [sent-400, score-0.27]

98 A note on the budgeted maximization on submodular functions. [sent-414, score-0.745]

99 Maximizing submodular set functions subject to multiple linear constraints. [sent-418, score-0.424]

100 Sentence compression as a component of a multi-document summarization system. [sent-501, score-0.267]


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