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CS 430 / INFO 430 Information Retrieval. Lecture 9 Latent Semantic Indexing. Course Administration. Latent Semantic Indexing. Objective Replace indexes that use sets of index terms by indexes that use concepts . Approach
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CS 430 / INFO 430 Information Retrieval Lecture 9 Latent Semantic Indexing
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.
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)
Example Query: "IDF in computer-based information look-up" Index terms for a document:access, document, retrieval, indexing How can we recognize that informationlook-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?
Technical Memo Example: Titles c1 Human machine interface for Lab ABC computer applications c2 A survey of user opinion of computer system response time c3 The EPS user interface management system c4 System and humansystem engineering testing of EPS c5 Relation of user-perceived responsetime to error measurement m1 The generation of random, binary, unordered trees m2 The intersection graph of paths in trees m3 Graph minors IV: Widths of trees and well-quasi-ordering m4 Graph minors: A survey
Technical Memo Example: Terms and Documents TermsDocuments c1 c2 c3 c4 c5 m1 m2 m3 m4 human 1 0 0 1 0 0 0 0 0 interface 1 0 1 0 0 0 0 0 0 computer 1 1 0 0 0 0 0 0 0 user 0 1 1 0 1 0 0 0 0 system 0 1 1 2 0 0 0 0 0 response 0 1 0 0 1 0 0 0 0 time 0 1 0 0 1 0 0 0 0 EPS 0 0 1 1 0 0 0 0 0 survey 0 1 0 0 0 0 0 0 1 trees 0 0 0 0 0 1 1 1 0 graph 0 0 0 0 0 0 1 1 1 minors 0 0 0 0 0 0 0 1 1
Technical Memo Example: Query Query: Find documents relevant to "human computer interaction" Simple Term Matching: Matches c1, c2, and c4 Misses c3 and c5
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.
Selection of Two-mode Factor Analysis Additional criterion: Computationally efficient O(N2k3) N is number of terms plus documents k is number of dimensions
The term vector space t3 The space has as many dimensions as there are terms in the word list. d1 d2 t2 t1
Figure 1 Latent concept vector space • term document query --- cosine > 0.9
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 T0, S0 and D0', such that: X = T0S0D0' T0 and D0 are the matrices of left and right singular vectors T0 and D0 have orthonormal columns S0 is the diagonal matrix of singular values
Dimensions of matrices t x d t x m m x m m x d S0 D0' X = T0 m is the rank of X< min(t, d)
Reduced Rank ~ ~ S0 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 S0 and the corresponding rows and columns of T0 and D0. 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 synonymyand recognizes dependence. ^ ^
Selection of singular values t x d t x k k x k k x d S D' ^ = X T k is the number of singular values chosen to represent the concepts in the set of documents. Usually, k« m.
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. ^ - - -
Calculation Similarities in the Concept Space Objective: Calculate similarities between terms, documents, and queries, using the matrices T, S, and D.
Mathematical Revision A is a p x q matrix B is a r x q matrix ai is the vector represented by row i of A bj is the vector represented by row j of B The inner product ai.bj is element i, j of AB' q r ith row of A q p B' jth row of B A
Comparing Two Terms ^ The dot product of two rows of X reflects the extent to which two terms have a similar pattern of occurrences. ^ ^ XX' = TSD'(TSD')' = TSD'DS'T' =TSS'T' Since D is orthonormal = TS(TS)' To calculate thei, jcell, take the dot product between the i and j rows ofTS Since S is diagonal, TS differs from T only by stretching the coordinate system
Comparing Two Documents ^ The dot product of two columns of X reflects the extent to which two columns have a similar pattern of occurrences. ^ ^ X'X = (TSD')'TSD' = DS(DS)' To calculate thei, jcell, take the dot product between the i and j columns ofDS. Since S is diagonal DS differs from D only by stretching the coordinate system
Comparing a Query and a Document A query can be expressed as a vector in the term-document vector space xq. xqi= 1 if term i is in the query and 0 otherwise. (Ignore query terms that are not in the term vector space.) Let pqj be the inner product of the queryxqwith document dj in the term-document vector space. pqj is the jth element in the product of xq'X. ^
Comparing a Query and a Document ^ X [pq1... pqj ... pqt] = [xq1 xq2 ... xqt] document dj is column j of X ^ inner product of query q with document dj query ^ pq' = xq'X = xq'TSD' = xq'T(DS)' similarity(q, dj) = cosine of angle is inner product divided by lengths of vectors pqj |xq| |dj|
Comparing a Query and a Document In the reading, the authors treat the query as a pseudo-document in the concept space dq: dq = xq'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 jth element of the product of: dqS and(DS)' This is the jth element of product of: xq'T (DS)' which is the same expression as before. Note that with their notation dq is a row vector.
Technical Memo Example: Query Terms Query xq human 1 interface 0 computer 0 user 0 system 1 response 0 time 0 EPS 0 survey 0 trees 1 graph 0 minors 0 Query: "humansystem interactions on trees" In term-document space, a query is represented by xq, a column vector with t elements. In concept space, a query is represented by dq, a row vector with k elements.
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.
