Computing Word Senses by Semantic Mirroring and Spectral Graph Partitioning
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1 Computing Word Senses by Semantic Mirroring and Spectral Graph Partitioning Masters Thesis by Martin Fagerlund Martin Fagerlund, Magnus Merkel, Lars Eldén, Lars Ahrenberg Departments of Mathematics and Computer Science, Linköping University ACL 2010 (LiU) ACL / 37
2 Outline 1 Semantic Mirrors 2 Graph and Spectral Theory Graph theory Spectral theory 3 The Computation of Word Senses 4 Evaluation and Results (LiU) ACL / 37
3 Semantic Mirrors (Dyvik) Use translations to extract information from a bilingual lexicon. Semantically closely related words tend to have strongly overlapping sets of translations. Words with a wide meaning tend to have a higher number of translations, than words with a more narrow meaning. (LiU) ACL / 37
4 Semantic Mirrors (LiU) ACL / 37
5 Semantic Mirrors (LiU) ACL / 37
6 Semantic Mirrors (LiU) ACL / 37
7 Semantic Mirrors (LiU) ACL / 37
8 Graph theory Connected graph with vertex set V and edge set E. (LiU) ACL / 37
9 Graph theory Disconnected graph. (LiU) ACL / 37
10 Graph theory The weight function. w(v i, v j ) = w(v j, v i ). w(v i, v j ) 0. (LiU) ACL / 37
11 Graph theory - Graphs and Matrices The adjacency matrix. { w(vi, v A(i, j) = j ), if v i and v j are adjacent, 0, otherwise. (LiU) ACL / 37
12 Graph theory - Graphs and Matrices The Laplacian. d(v i ) = j w(v i, v j ). L = D A. D(i, j) = { d(vi ), if i = j, 0, otherwise. (LiU) ACL / 37
13 Graph theory - Graphs and Matrices The Laplacian. d(v i ) = j w(v i, v j ). L = D A. D(i, j) = { d(vi ), if i = j, 0, otherwise. (LiU) ACL / 37
14 Graph theory - Graphs and Matrices The normalised Laplacian. L = D 1/2 LD 1/2. 1 w(v i,v i ) d(v i ), if i = j and d(v i) 0, L = w(v i,v j ) d(vi )d(v j ), if v i and v j are adjacent, 0, otherwise. (LiU) ACL / 37
15 Spectral graph partitioning Compute the eigenvalues and the eigenvectors 0 = λ 0 λ 1... λ n 1, ū 0,..., ū n 1 of L. λ 1 is called the Fiedler value, and ū 1 is called the Fiedler vector. (LiU) ACL / 37
16 Spectral graph partitioning Let ū = (u 1,..., u n ) be the Fiedler vector. Divide the vertices into S and S using a splitting value s: v i S if u i > s v i S if u i s. A few ways of choosing s: bisection: s is the median of {u 1,..., u n }. sign cut: s =0. gap cut: s is a value in the largest gap in the sorted Fiedler vector. (LiU) ACL / 37
17 Spectral graph partitioning - An example (LiU) ACL / 37
18 Spectral graph partitioning - An example (LiU) ACL / 37
19 Spectral graph partitioning Measures of the partitioning. Let d(s) = v i S d(v i ), and Conductance, E(S, S) = v i S,v j S w(v i, v j ). E(S, S) φ(s) = d(v ) d(s)d( S), (LiU) ACL / 37
20 Spectral graph partitioning Measures of the partitioning. Let d(s) = v i S d(v i ), and E(S, S) = w(v i, v j ). v i S,v j S Sparsity, sp(s) = E(S, S) min(d(s), d( S)) (LiU) ACL / 37
21 Spectral graph partitioning The conductance of a graph The sparsity of a graph φ G = min φ(s) S sp G = min sp(s) S (LiU) ACL / 37
22 Spectral graph partitioning The Cheeger inequalities 2φ G > λ 1 φ2 G 8. 2sp G λ 1 sp2 G 2. (LiU) ACL / 37
23 Separating word senses - Translation English-Swedish lexicon of adjectives. Words into two lists. Translation matrix B. Rows correspond to English words. Columns correspond to Swedish words. (LiU) ACL / 37
24 Separating word senses - Translation English word j defines a vector ē j by { 1 if i = j, ē j (i) = 0 if i j. B T ē j gives translations from English to Swedish. (LiU) ACL / 37
25 Separating word senses - Computation Start with an English word, called eng1. BB T ē eng1. eng2 swe1 eng3 eng1 swe2 eng1 swe3 eng4 eng5 (LiU) ACL / 37
26 Separating word senses - Computation Start with an English word, called eng1. BB T ē eng1. eng2 swe1 eng3 eng1 swe2 eng1 swe3 eng4 eng5 (LiU) ACL / 37
27 Separating word senses - Computation Replace the row in B corresponding to eng1, with an all-zero row. Call this new matrix B mod1. B mod1 B T ē eng1. eng2 swe1 eng3 eng1 swe2 eng1 swe3 eng4 eng5 (LiU) ACL / 37
28 Separating word senses - Computation Replace the row in B corresponding to eng1, with an all-zero row. Call this new matrix B mod1. B mod1 B T ē eng1. eng2 swe1 eng3 eng1 swe2 eng1 swe3 eng4 eng5 (LiU) ACL / 37
29 Separating word senses - Computation Let B mod2 be the matrix B, with columns corresponding to the Swedish words swe1,...,swe3, replaced with all-zero columns. Then B mod2 B T mod2 E (LiU) ACL / 37
30 Separating word senses - Computation swe4 eng6 eng2 swe5 eng2 swe1 eng3 swe1 eng3 eng1 swe2 eng1 swe2 eng1 swe3 eng4 swe3 eng4 eng5 swe6 eng5 swe7 eng7 (LiU) ACL / 37
31 Separating word senses - Computation If in B mod2 B T mod2 E keeping only the rows corresponding to eng2,...eng5, we get a symmetric matrix A. A = (LiU) ACL / 37
32 Separating word senses - Computation (LiU) ACL / 37
33 Separating word senses - Computation Create L, and compute the Fiedler vector. Sort the Fiedler vector, and make n 1 cuts. For each cut, compute the conductance and choose the the cut that produce the lowest value. Partition the graph. (LiU) ACL / 37
34 Separating word senses - Computation Continue and partition the largest of the subgraphs obtained. End the iteration when a stopping criterion is fulfilled. (LiU) ACL / 37
35 Separating word senses - Example 1 Global absolute full-scale round aggregate intact teetotal all-out integral total clear integrate unbroken complete international universal entire mondial utter full one-piece whole full-length outright worldwide (LiU) ACL / 37
36 Separating word senses - Example 1 Global intact absolute integrate aggregate whole utter unbroken full-scale full-length complete integral one-piece all-out clear teetotal round full outright entire total worldwide mondial universal international (LiU) ACL / 37
37 Separating word senses - Example 2 Visiting adventitious alien exotic foreign outlandish strange unaccustomed uncouth unfamiliar ungenial (LiU) ACL / 37
38 Separating word senses - Example 2 Visiting alien foreign outlandish exotic adventitious ungenial unaccustomed unfamiliar strange uncouth (LiU) ACL / 37
39 Evaluation and Results Evaluation Results 10 random words with at least 8 vertices in the graph were evaluated. Both conductance and sparsity were used as a measure. The sense groups are not always completely synonymous with the seed word. The words are often consistent within the sense groups. (LiU) ACL / 37
40 Evaluation and Results Evaluation Results 10 random words with at least 8 vertices in the graph were evaluated. Both conductance and sparsity were used as a measure. The sense groups are not always completely synonymous with the seed word. The words are often consistent within the sense groups. (LiU) ACL / 37
41 Conclusions and future work Conclusion Our preliminary results indicate that graph partitioning-based semantic mirroring can be developed into an automatic method for computing word senses. Future work automatic treatment of homonyms Tests on other dictionaries, languages Systematic evaluation of the method To get a copy of Martin Fagerlund s Masters thesis, send me or him a message. (LiU) ACL / 37
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