1 CS 430: Information Discovery Lecture 11 Latent Semantic Indexing

2 Course Administration Comments on Assignment 1. Office hours. See correction on web site.

3 Reading Scott Deerwester, Susan T. Dumais, George W. Furnas, Thomas K. Landauer, Richard Harshman, "Indexing by latent semantic analysis". Journal of the American Society for Information Science, Volume 41, Issue 6, bin/issuetoc?ID=

4 Latent Semantic Indexing Objective Replace indexes that use sets of index terms by indexes that use concepts. Approach Map the index term vector space into a lower dimensional space, using singular value decomposition.

5 t1t1 t2t2 t3t3 d1d1 d2d2  The space has as many dimensions as there are terms in the word list. The index term vector space

6 Deficiencies with Conventional Automatic Indexing Synonymy: Various words and phrases refer to the same concept (lowers recall). Polysemy: Individual words have more than one meaning (lowers precision) Independence: No significance is given to two terms that frequently appear together

7 Example Query: "IDF in computer-based information look-up" Index terms for a document: access, document, retrieval, indexing How can we recognize that information look-up is related to retrieval and indexing? Conversely, if information has many different contexts in the set of documents, how can we discover that it is an unhelpful term for retrieval?

8 Models of Semantic Similarity Proximity models: Put similar items together in some space or structure Clustering (hierarchical, partition, overlapping). Documents are considered close to the extent that they contain the same terms. Most then arrange the documents into a hierarchy based on distances between documents. [Covered later in course.] Factor analysis based on matrix of similarities between documents (single mode). Two-mode proximity methods. Start with rectangular matrix and construct explicit representations of both row and column objects.

9 Selection of Two-mode Factor Analysis Additional criterion: Computationally efficient O(N 2 k 3 ) N is number of terms plus documents k is number of dimensions

10 Technical Memo Example: Titles c1Human machine interface for Lab ABC computer applications c2A survey of user opinion of computer system response time c3The EPS user interface management system c4System and human system engineering testing of EPS c5Relation of user-perceived response time to error measurement m1The generation of random, binary, unordered trees m2The intersection graph of paths in trees m3Graph minors IV: Widths of trees and well-quasi-ordering m4Graph minors: A survey

11 Technical Memo Example: Terms and Documents Terms Documents c1c2c3c4c5m1m2m3m4 human interface computer user system response time EPS survey trees graph minors

12 Technical Memo Example: Query Query: Find documents relevant to "human computer interaction" Simple Term Matching: Matches c1, c2, and c3 Misses c4 and c5

13 Figure 1 term document query --- cosine > 0.9

14 Mathematical concepts Singular Value Decomposition Define X as the term-document matrix, with t rows (number of index terms) and n columns (number of documents). There exist matrices T, S and D', such that: X = T 0 S 0 D 0 ' T 0 and D 0 are the matrices of left and right singular vectors T 0 and D 0 have orthonormal columns S 0 is the diagonal matrix of singular values

15 Dimensions of matrices X= T0T0 D0'D0'S0S0 t x dt x mm x dm x m m is the rank of X < min(t, d)

16 Reduced Rank Diagonal elements of S 0 are positive and decreasing in magnitude. Keep the first k and set the others to zero. Delete the zero rows and columns of S 0 and the corresponding rows and columns of T 0 and D 0. This gives: X X = TSD' Interpretation If value of k is selected well, expectation is that X retains the semantic information from X, but eliminates noise from synonymy and recognizes dependence. ~ ~ ^ ^

17 Selection of singular values X = t x dt x kk x dk x k k is the number of singular values chosen to represent the concepts in the set of documents. Usually, k « m. T SD' ^

18 Comparing Two Terms XX' = TSD'(TSD')' = TSD'DS'T' = TSS'T Since D is orthonormal = TS(TS)' To calculate the i, j cell, take the dot product between the i and j rows of TS Since S is diagonal, TS differs from T only by stretching the coordinate system ^ ^ The dot product of two rows of X reflects the extent to which two terms have a similar pattern of occurrences. ^

19 Comparing Two Documents X'X = (TSD')'TSD' = DS(DS)' To calculate the i, j cell, take the dot product between the i and j columns of DS. Since S is diagonal DS differs from D only by stretching the coordinate system ^ ^ The dot product of two columns of X reflects the extent to which two columns have a similar pattern of occurrences. ^

20 Comparing a Term and a Document Comparison between a term and a document is the value of an individual cell of X. X = TSD' = TS(DS)' where S is a diagonal matrix whose values are the square root of the corresponding elements of S. ^ - - -

21 Technical Memo Example: Query Terms Query x q human1 interface0 computer0 user0 system1 response0 time0 EPS0 survey0 trees1 graph0 minors0 Query: "human system interactions on trees" In term-document space, a query is represented by x q, a t x 1 vector. In concept space, a query is represented by d q, a 1 x k vector.

22 Query Suggested form of d q is: d q = x q 'TS -1 Example of use. To compare a query against document i, take the i th element of the product of DS and d q S, which is the i th element of product of DS and x q 'T. Note that is a d q row vector.

23 Query Let x q be the vector of terms for a query q. In the reduced dimensional space, q, is represented by a pseudo-document, d q, at the centroid of the corresponding term points, with appropriate rescaling of the axes. d q = x q 'TS -1

24 Experimental Results Deerwester, et al. tried latent semantic indexing on two test collections, MED and CISI, where queries and relevant judgments available. Documents were full text of title and abstract. Stop list of 439 words (SMART); no stemming, etc. Comparison with: (a) simple term matching, (b) SMART, (c) Voorhees method.

25 Experimental Results: 100 Factors

26 Experimental Results: Number of Factors