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Paper: Indexing by Latent Semantic Analysis

Paper: Indexing by Latent Semantic Analysis. for course cs630 presented by: Haiyan Qiao. Problem Introduction. Traditional term-matching method doesn’t work well in information retrieval We want to capture the concepts instead of words. Concepts are reflected in the words. However,

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Paper: Indexing by Latent Semantic Analysis

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  1. Paper: Indexing by Latent Semantic Analysis for course cs630 presented by: Haiyan Qiao

  2. Problem Introduction • Traditional term-matching method doesn’t work well in information retrieval • We want to capture the concepts instead of words. Concepts are reflected in the words. However, • One term may have multiple meaning • Different terms may have the same meaning.

  3. LSI (Latent Semantic Analysis) • LSI approach tries to overcome the deficiencies of term-matching retrieval by treating the unreliability of observed term-document association data as a statistical problem. • The goal is to find effective models to represent the relationship between terms and documents. Hence a set of terms, which is by itself incomplete and unreliable, will be replaced by some set of entities which are more reliable indicants.

  4. SVD (Singular Value Decomposition) • How to learn the concepts from data? • SVD is applied to derive the latent semantic structure model. • What is SVD? http://kwon3d.com/theory/jkinem/svd.html http://mathworld.wolfram.com/SingularValueDecomposition.html http://www.cs.ut.ee/~toomas_l/linalg/lin2/node13.html#SECTION00013200000000000000

  5. SVD cont’ • SVD of the term-by-document matrix X: • If the singular values of S0 are ordered by size, we only keep the first k largest values and get a reduced model: • doesn’t exactly match X and it gets closer as more and more singular values are kept • This is what we want. We don’t want perfect fit since we think some of 0’s in X should be 1 and vice versa. • It reflects the major associative patterns in the data, and ignores the smaller, less important influence and noise.

  6. Fundamental Comparison Quantities from the SVD Model • Comparing Two Terms: the dot product between two row vectors of reflects the extent to which two terms have a similar pattern of occurrence across the set of document. • Comparing Two Documents: dot product between two column vectors of • Comparing a Term and a Document

  7. Example -Technical Memo • Query: human-computer interaction • Dataset: c1 Human machine interface for Lab ABC computer application c2 A survey of user opinion of computersystemresponsetime c3 The EPS user interface management system c4 System and humansystem engineering testing of EPS c5 Relations of user-perceived responsetime to error measurement m1 The generation of random, binary, unordered trees m2 The intersection graph of paths in trees m3 Graphminors IV: Widths of trees and well-quasi-ordering m4 Graphminors: A survey

  8. Example cont’ % 12-term by 9-document matrix >> X=[ 1 0 0 1 0 0 0 0 0; 1 0 1 0 0 0 0 0 0; 1 1 0 0 0 0 0 0 0; 0 1 1 0 1 0 0 0 0; 0 1 1 2 0 0 0 0 0 0 1 0 0 1 0 0 0 0; 0 1 0 0 1 0 0 0 0; 0 0 1 1 0 0 0 0 0; 0 1 0 0 0 0 0 0 1; 0 0 0 0 0 1 1 1 0; 0 0 0 0 0 0 1 1 1; 0 0 0 0 0 0 0 1 1;];

  9. Example cont’ % X=T0*S0*D0', T0 and D0 have orthonormal columns and So is diagonal % T0 is the matrix of eigenvectors of the square symmetric matrix XX' % D0 is the matrix of eigenvectors of X’X % S0 is the matrix of eigenvalues in both cases >> [T0, S0] = eig(X*X'); >> T0 T0 =   0.1561 -0.2700 0.1250 -0.4067 -0.0605 -0.5227 -0.3410 -0.1063 -0.4148 0.2890 -0.1132 0.2214 0.1516 0.4921 -0.1586 -0.1089 -0.0099 0.0704 0.4959 0.2818 -0.5522 0.1350 -0.0721 0.1976 -0.3077 -0.2221 0.0336 0.4924 0.0623 0.3022 -0.2550 -0.1068 -0.5950 -0.1644 0.0432 0.2405 0.3123 -0.5400 0.2500 0.0123 -0.0004 -0.0029 0.3848 0.3317 0.0991 -0.3378 0.0571 0.4036 0.3077 0.2221 -0.0336 0.2707 0.0343 0.1658 -0.2065 -0.1590 0.3335 0.3611 -0.1673 0.6445 -0.2602 0.5134 0.5307 -0.0539 -0.0161 -0.2829 -0.1697 0.0803 0.0738 -0.4260 0.1072 0.2650 -0.0521 0.0266 -0.7807 -0.0539 -0.0161 -0.2829 -0.1697 0.0803 0.0738 -0.4260 0.1072 0.2650 -0.7716 -0.1742 -0.0578 -0.1653 -0.0190 -0.0330 0.2722 0.1148 0.1881 0.3303 -0.1413 0.3008 0.0000 0.0000 0.0000 -0.5794 -0.0363 0.4669 0.0809 -0.5372 -0.0324 -0.1776 0.2736 0.2059 0.0000 0.0000 0.0000 -0.2254 0.2546 0.2883 -0.3921 0.5942 0.0248 0.2311 0.4902 0.0127 -0.0000 -0.0000 -0.0000 0.2320 -0.6811 -0.1596 0.1149 -0.0683 0.0007 0.2231 0.6228 0.0361 0.0000 -0.0000 0.0000 0.1825 0.6784 -0.3395 0.2773 -0.3005 -0.0087 0.1411 0.4505 0.0318

  10. Example cont’ >> [D0, S0] = eig(X'*X); >> D0 D0 =    0.0637 0.0144 -0.1773 0.0766 -0.0457 -0.9498 0.1103 -0.0559 0.1974 -0.2428 -0.0493 0.4330 0.2565 0.2063 -0.0286 -0.4973 0.1656 0.6060 -0.0241 -0.0088 0.2369 -0.7244 -0.3783 0.0416 0.2076 -0.1273 0.4629 0.0842 0.0195 -0.2648 0.3689 0.2056 0.2677 0.5699 -0.2318 0.5421 0.2624 0.0583 -0.6723 -0.0348 -0.3272 0.1500 -0.5054 0.1068 0.2795 0.6198 -0.4545 0.3408 0.3002 -0.3948 0.0151 0.0982 0.1928 0.0038 -0.0180 0.7615 0.1522 0.2122 -0.3495 0.0155 0.1930 0.4379 0.0146 -0.5199 -0.4496 -0.2491 -0.0001 -0.1498 0.0102 0.2529 0.6151 0.0241 0.4535 0.0696 -0.0380 -0.3622 0.6020 -0.0246 0.0793 0.5299 0.0820

  11. Example cont’ >> S0=eig(X'*X) >> S0=S0.^0.5 S0 =   0.3637 0.5601 0.8459 1.3064 1.5048 1.6445 2.3539 2.5417 3.3409 % We only keep the largest two singular values % and the corresponding columns from the T and D

  12. Example cont’ >> T=[0.2214 -0.1132; 0.1976 -0.0721; 0.2405 0.0432; 0.4036 0.0571; 0.6445 -0.1673; 0.2650 0.1072; 0.2650 0.1072; 0.3008 -0.1413; 0.2059 0.2736; 0.0127 0.4902; 0.0361 0.6228; 0.0318 0.4505;]; >> S = [ 3.3409 0; 0 2.5417 ]; >> D’ =[0.1974 0.6060 0.4629 0.5421 0.2795 0.0038 0.0146 0.0241 0.0820; -0.0559 0.1656 -0.1273 -0.2318 0.1068 0.1928 0.4379 0.6151 0.5299;] >> T*S*D’ 0.1621 0.4006 0.3790 0.4677 0.1760 -0.0527 0.1406 0.3697 0.3289 0.4004 0.1649 -0.0328 0.1525 0.5051 0.3580 0.4101 0.2363 0.0242 0.2581 0.8412 0.6057 0.6973 0.3924 0.0331 0.4488 1.2344 1.0509 1.2658 0.5564 -0.0738 0.1595 0.5816 0.3751 0.4168 0.2766 0.0559 0.1595 0.5816 0.3751 0.4168 0.2766 0.0559 0.2185 0.5495 0.5109 0.6280 0.2425 -0.0654 0.0969 0.5320 0.2299 0.2117 0.2665 0.1367 -0.0613 0.2320 -0.1390 -0.2658 0.1449 0.2404 -0.0647 0.3352 -0.1457 -0.3016 0.2028 0.3057 -0.0430 0.2540 -0.0966 -0.2078 0.1520 0.2212

  13. Summary • What is the common and difference between PCA and SVD? • Both are related to standard eigenvalue-eigenvector, to remove noise or correlation and get the most important info. • PCA is on covariance matrix and SVD works on original matrix.

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