1 CS 430 / INFO 430 Information Retrieval Lecture 9 Latent Semantic Indexing

2 Course Administration

3 Latent Semantic Indexing Objective Replace indexes that use sets of index terms by indexes that use concepts. Approach Map the term vector space into a lower dimensional space, using singular value decomposition. Each dimension in the new space corresponds to a latent concept in the original data.

4 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 Latent semantic indexing addresses the first of these (synonymy), and the third (dependence)

5 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?

6 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

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

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

9 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.

10 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

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

12 Figure 1 term document query --- cosine > 0.9 Latent concept vector space

13 Mathematical concepts Define X as the term-document matrix, with t rows (number of index terms) and d columns (number of documents). Singular Value Decomposition For any matrix X, with t rows and d columns, there exist matrices T 0, S 0 and D 0 ', 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

14 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)

15 Reduced Rank S 0 can be chosen so that the diagonal elements 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. ~ ~ ^ ^

16 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' ^

17 Comparing a Term and a Document An individual cell of X is the number of occurrences of term i in document j. X = TSD' = TS(DS)' where S is a diagonal matrix whose values are the square root of the corresponding elements of S. ^ ^

18 Calculation Similarities in the Concept Space Objective: Calculate similarities between terms, documents, and queries, using the matrices T, S, and D.

19 Mathematical Revision A is a p x q matrix B is a r x q matrix a i is the vector represented by row i of A b j is the vector represented by row j of B The inner product a i.b j is element i, j of AB' p q q r A B' i th row of A j th row of B

20 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. ^

21 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. ^

22 Comparing a Query and a Document A query can be expressed as a vector in the term- document vector space x q. x qi = 1 if term i is in the query and 0 otherwise. (Ignore query terms that are not in the term vector space.) Let p qj be the inner product of the query x q with document d j in the term-document vector space. p qj is the j th element in the product of x q 'X. ^

23 Comparing a Query and a Document [p q1... p qj... p qt ] = [x q1 x q2... x qt ] ^ X inner product of query q with document d j query document d j is column j of X ^ p q ' = x q 'X = x q 'TSD' = x q 'T(DS)' similarity(q, d j ) = ^ p qj |x q | |d j | cosine of angle is inner product divided by lengths of vectors

24 Comparing a Query and a Document In the reading, the authors treat the query as a pseudo- document in the concept space d q : d q = x q 'TS -1 [Note that S -1 stretches the vector] To compare a query against document j, they extend the method used to compare document i with document j. Take the j th element of the product of: d q S and (DS)' This is the j th element of product of: x q 'T (DS)' which is the same expression as before. Note that with their notation d q is a row vector.

25 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 column vector with t elements. In concept space, a query is represented by d q, a row vector with k elements.

26 Experimental Results Deerwester, et al. tried latent semantic indexing on two test collections, MED and CISI, where queries and relevant judgments were 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.

27 Experimental Results: 100 Factors

28 Experimental Results: Number of Factors