Latent semantic indexing

This is a method discussed in Information Extraction 10-707 in Fall 2010.

Latent Semantic Indexing (LSI) is an indexing and retrieval method that uses a mathematical technique called Singular Value Decomposition (SVD) to identify patterns in the relationships between the terms and concepts contained in an unstructured collection of text. LSI is based on the principle that words that are used in the same contexts tend to have similar meanings. A key feature of LSI is its ability to extract the conceptual content of a body of text by establishing associations between those terms that occur in similar contexts.<ref>Deerwester, S., et al, Improving Information Retrieval with Latent Semantic Indexing, Proceedings of the 51st Annual Meeting of the American Society for Information Science 25, 1988, pp. 36–40.</ref>

LSI is also an application of Correspondence Analysis, a multivariate statistical technique developed by Jean-Paul Benzécri<ref>Template:Cite book</ref> in the early 1970s, to a Contingency Table built from word counts in documents.

Called Latent Semantic Indexing because of its ability to correlate semantically related terms that are latent in a collection of text, it was first applied to text at Bell Laboratories in the late 1980s. The method, also called Latent Semantic Analysis (LSA), uncovers the underlying latent semantic structure in the usage of words in a body of text and how it can be used to extract the meaning of the text in response to user queries, commonly referred to as concept searches. Queries, or concept searches, against a set of documents that have undergone LSI will return results that are conceptually similar in meaning to the search criteria even if the results don’t share a specific word or words with the search criteria.

Mathematics of LSI

LSI uses common linear algebra techniques to learn the conceptual correlations in a collection of text. In general, the process involves constructing a weighted term-document matrix, performing a Singular Value Decomposition on the matrix, and using the matrix to identify the concepts contained in the text.

Term Document Matrix

LSI begins by constructing a term-document matrix, $A$, to identify the occurrences of the $m$ unique terms within a collection of $n$ documents. In a term-document matrix, each term is represented by a row, and each document is represented by a column, with each matrix cell, $a_{ij}$, initially representing the number of times the associated term appears in the indicated document, $\mathrm {tf_{ij}}$. This matrix is usually very large and very sparse.

Once a term-document matrix is constructed, local and global weighting functions can be applied to it to condition the data. The weighting functions transform each cell, $a_{ij}$ of $A$, to be the product of a local term weight, $l_{ij}$, which describes the relative frequency of a term in a document, and a global weight, $g_{i}$, which describes the relative frequency of the term within the entire collection of documents.

Some common local weighting functions <ref> Berry, M. W., and Browne, M., Understanding Search Engines: Mathematical Modeling and Text Retrieval, Society for Industrial and Applied Mathematics, Philadelphia, (2005).</ref> are defined in the following table.

Binary		$l_{ij}=1$ if the term exists in the document, or else $0$
TermFrequency		$l_{ij}=\mathrm {tf} _{ij}$, the number of occurrences of term $i$ in document $j$
Log		$l_{ij}=\log(\mathrm {tf} _{ij}+1)$
Augnorm		$l_{ij}={\frac {{\Big (}{\frac {\mathrm {tf} _{ij}}{\max _{i}(\mathrm {tf} _{ij})}}{\Big )}+1}{2}}$

Some common global weighting functions are defined in the following table.

Binary		$g_{i}=1$
Normal		$g_{i}={\frac {1}{\sqrt {\sum _{j}\mathrm {tf} _{ij}^{2}}}}$
GfIdf		$g_{i}=\mathrm {gf} _{i}/\mathrm {df} _{i}$, where $\mathrm {gf} _{i}$ is the total number of times term $i$ occurs in the whole collection, and $\mathrm {df} _{i}$ is the number of documents in which term $i$ occurs.
Idf		$g_{i}=1+\log _{2}{\frac {n}{\mathrm {df} _{i}}}$
Entropy		$g_{i}=1+\sum _{j}{\frac {p_{ij}\log p_{ij}}{\log n}}$, where $p_{ij}={\frac {\mathrm {tf} _{ij}}{\mathrm {gf} _{i}}}$

Empirical studies with LSI report that the Log Entropy weighting functions work well, in practice, with many data setsTemplate:Citation needed. In other words, each entry $a_{ij}$ of $A$ is computed as:

$g_{i}=1+\sum _{j}{\frac {p_{ij}\log p_{ij}}{\log n}}$

$a_{ij}=g_{i}\ \log(\mathrm {tf} _{ij}+1)$

Rank-Reduced Singular Value Decomposition

A rank-reduced, Singular Value Decomposition is performed on the matrix to determine patterns in the relationships between the terms and concepts contained in the text. The SVD forms the foundation for LSI.<ref>Berry, Michael W., Dumais, Susan T., O'Brien, Gavin W., Using Linear Algebra for Intelligent Information Retrieval, December 1994, SIAM Review 37:4 (1995), pp. 573–595.</ref> It computes the term and document vector spaces by transforming the single term-frequency matrix, A, into three other matrices— a term-concept vector matrix, T, a singular values matrix, S, and a concept-document vector matrix, D, which satisfy the following relations:

A = TSD^T

T^TT = D^T D = I_r    TT^T = I_m    DD^T = I_n

S_1,1 ≥ S_2,2 ≥...≥ S_r,r > 0    S_i,j = 0 where i ≠ j

In the formula, A, is the supplied m by n weighted matrix of term frequencies in a collection of text where m is the number of unique terms, and n is the number of documents. T is a computed m by r matrix of term vectors where r is the rank of A—a measure of its unique dimensions ≤ min(m,n). S is a computed r by r diagonal matrix of decreasing singular values, and D is a computed n by r matrix of document vectors.

The LSI modification to a standard SVD is to reduce the rank or truncate the singular value matrix S to size k « r, typically on the order of a k in the range of 100 to 300 dimensions, effectively reducing the term and document vector matrix sizes to m by k and n by k respectively. The SVD operation, along with this reduction, has the effect of preserving the most important semantic information in the text while reducing noise and other undesirable artifacts of the original space of A. This reduced set of matrices is often denoted with a modified formula such as:

A ≈ A_k = T_k S_k D_k^T

Efficient LSI algorithms only compute the first k singular values and term and document vectors as opposed to computing a full SVD and then truncating it.

Querying and Augmenting LSI Vector Spaces

The computed T_k and D_k matrices define the term and document vector spaces, which with the computed singular values, S_k, embody the conceptual information derived from the document collection. The similarity of terms or documents within these spaces is a factor of how close they are to each other in these spaces, typically computed as a function of the angle between the corresponding vectors.

The same steps are used to locate the vectors representing the text of queries and new documents within the document space of an existing LSI index. By a simple transformation of the A = T S D^T equation into the equivalent D = A^T T S⁻¹ equation, a new vector, d, for a query or for a new document can be created by computing a new column in A and then multiplying the new column by T S⁻¹. The new column in A is computed using the originally derived global term weights and applying the same local weighting function to the terms in the query or in the new document.

A drawback to computing vectors in this way, when adding new searchable documents, is that terms that were not known during the SVD phase for the original index are ignored. These terms will have no impact on the global weights and learned correlations derived from the original collection of text. However, the computed vectors for the new text are still very relevant for similarity comparisons with all other document vectors.

The process of augmenting the document vector spaces for an LSI index with new documents in this manner is called folding-in. Although the folding-in process does not account for the new semantic content of the new text, adding a substantial number of documents in this way will still provide good results for queries as long as the terms and concepts they contain are well represented within the LSI index to which they are being added. When the terms and concepts of a new set of documents need to be included in an LSI index, the term-document matrix, and the SVD, must either be recomputed or an incremental update method (such as the one described in <ref name="brand2006">Template:Cite journal</ref>) be used.