# Latent semantic indexing

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.[1]

LSI is also an application of correspondence analysis, a multivariate statistical technique developed by Jean-Paul Benzécri[2] 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 Bellcore 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.

## Benefits of LSI

LSI overcomes two of the most problematic constraints of Boolean keyword queries: multiple words that have similar meanings (synonymy) and words that have more than one meaning (polysemy). Synonymy is often the cause of mismatches in the vocabulary used by the authors of documents and the users of information retrieval systems.[3][4] As a result, Boolean or keyword queries often return irrelevant results and misinformation that is relevant.

LSI is also used to perform automated document categorization. In fact, several experiments have demonstrated that there are a number of correlations between the way LSI and humans process and categorize text.[5] Document categorization is the assignment of documents to one or more predefined categories based on their similarity to the conceptual content of the categories.[6] LSI uses example documents to establish the conceptual basis for each category. During categorization processing, the concepts contained in the documents being categorized are compared to the concepts contained in the example items, and a category (or categories) is assigned to the documents based on the similarities between the concepts they contain and the concepts that are contained in the example documents.

Dynamic clustering based on the conceptual content of documents can also be accomplished using LSI. Clustering is a way to group documents based on their conceptual similarity to each other without using example documents to establish the conceptual basis for each cluster. This is very useful when dealing with an unknown collection of unstructured text.

Because it uses a strictly mathematical approach, LSI is inherently independent of language. This enables LSI to elicit the semantic content of information written in any language without requiring the use of auxiliary structures, such as dictionaries and thesauri. LSI can also perform cross-linguistic concept searching and example-based categorization. For example, queries can be made in one language, such as English, and conceptually similar results will be returned even if they are composed of an entirely different language or of multiple languages.

LSI is not restricted to working only with words. It can also process arbitrary character strings. Any object that can be expressed as text can be represented in an LSI vector space.[7] For example, tests with MEDLINE abstracts have shown that LSI is able to effectively classify genes based on conceptual modeling of the biological information contained in the titles and abstracts of the MEDLINE citations.[8]

LSI automatically adapts to new and changing terminology, and has been shown to be very tolerant of noise (i.e., misspelled words, typographical errors, unreadable characters, etc.).[9] This is especially important for applications using text derived from Optical Character Recognition (OCR) and speech-to-text conversion. LSI also deals effectively with sparse, ambiguous, and contradictory data.

Text does not need to be in sentence form for LSI to be effective. It can work with lists, free-form notes, email, Web-based content, etc. As long as a collection of text contains multiple terms, LSI can be used to identify patterns in the relationships between the important terms and concepts contained in the text.

LSI has proven to be a useful solution to a number of conceptual matching problems.[10][11] The technique has been shown to capture key relationship information, including causal, goal-oriented, and taxonomic information.[12]

## LSI timeline

Mid-1960s – Factor analysis technique first described and tested (H. Borko and M. Bernick)

1988 – Seminal paper on LSI technique published (Deerwester et al.)

1989 – Original patent granted (Deerwester et al.)

1992 – First use of LSI to assign articles to reviewers[13] (Dumais and Nielsen)

1994 – Patent granted for the cross-lingual application of LSI (Landauer et al.)

1995 – First use of LSI for grading essays (Foltz, et al., Landauer et al.)

1999 – First implementation of LSI technology for intelligence community for analyzing unstructured text (SAIC).

2002 – LSI-based product offering to intelligence-based government agencies (SAIC)

2005 – First vertical-specific application – publishing – EDB (EBSCO, Content Analyst Company)

## 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 [14] 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 $g_i = \log_2 \frac{n}{1+ \mathrm{df}_i}$ 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 sets.[15] 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.[16] It computes the term and document vector spaces by transforming the single term-frequency matrix, $A$, into three other matrices— an m by r term-concept vector matrix $T$, an r by r singular values matrix $S$, and a n by r concept-document vector matrix, $D$, which satisfy the following relations:

$A = TSD^T$

$T^T T = I_r \quad D^T D = I_r$

$S_{1,1} \geq S_{2,2} \geq \ldots \geq S_{r,r} > 0 \quad S_{i,j} = 0 \; \text{where} \; i \neq j$

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 ≈ Ak = Tk Sk DkT

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.

Note that this rank reduction is essentially the same as doing Principal Component Analysis (PCA) on the matrix A, except that PCA subtracts off the means. PCA loses the sparseness of the A matrix, which can make it infeasible for large lexicons.

## Querying and augmenting LSI vector spaces

The computed Tk and Dk matrices define the term and document vector spaces, which with the computed singular values, Sk, 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 DT equation into the equivalent D = AT 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, either the term-document matrix, and the SVD, must be recomputed or an incremental update method (such as the one described in [17]) be used.

It is generally acknowledged that the ability to work with text on a semantic basis is essential to modern information retrieval systems. As a result, the use of LSI has significantly expanded in recent years as earlier challenges in scalability and performance have been overcome.

LSI is being used in a variety of information retrieval and text processing applications, although its primary application has been for concept searching and automated document categorization.[18] Below are some other ways in which LSI is being used:

• Information discovery[19] (eDiscovery, Government/Intelligence community, Publishing)
• Automated document classification (eDiscovery, Government/Intelligence community, Publishing)[20]
• Text summarization[21] (eDiscovery, Publishing)
• Relationship discovery[22] (Government, Intelligence community, Social Networking)
• Automatic generation of link charts of individuals and organizations[23] (Government, Intelligence community)
• Matching technical papers and grants with reviewers[24] (Government)
• Online customer support[25] (Customer Management)
• Determining document authorship[26] (Education)
• Automatic keyword annotation of images[27]
• Understanding software source code[28] (Software Engineering)
• Information visualization[30]
• Essay scoring[31] (Education)
• Literature-based discovery[32]

LSI is increasingly being used for electronic document discovery (eDiscovery) to help enterprises prepare for litigation. In eDiscovery, the ability to cluster, categorize, and search large collections of unstructured text on a conceptual basis is essential. Concept-based searching using LSI has been applied to the eDiscovery process by leading providers as early as 2003.[33]

## Challenges to LSI

Early challenges to LSI focused on scalability and performance. LSI requires relatively high computational performance and memory in comparison to other information retrieval techniques.[34] However, with the implementation of modern high-speed processors and the availability of inexpensive memory, these considerations have been largely overcome. Real-world applications involving more than 30 million documents that were fully processed through the matrix and SVD computations are not uncommon in some LSI applications. A fully scalable (unlimited number of documents, online training) implementation of LSI is contained in the open source gensim software package.[35]

Another challenge to LSI has been the alleged difficulty in determining the optimal number of dimensions to use for performing the SVD. As a general rule, fewer dimensions allow for broader comparisons of the concepts contained in a collection of text, while a higher number of dimensions enable more specific (or more relevant) comparisons of concepts. The actual number of dimensions that can be used is limited by the number of documents in the collection. Research has demonstrated that around 300 dimensions will usually provide the best results with moderate-sized document collections (hundreds of thousands of documents) and perhaps 400 dimensions for larger document collections (millions of documents).[36] However, recent studies indicate that 50-1000 dimensions are suitable depending on the size and nature of the document collection.[37]

Checking the amount of variance in the data after computing the SVD can be used to determine the optimal number of dimensions to retain. The variance contained in the data can be viewed by plotting the singular values (S) in a scree plot. Some LSI practitioners select the dimensionality associated with the knee of the curve as the cut-off point for the number of dimensions to retain. Others argue that some quantity of the variance must be retained, and the amount of variance in the data should dictate the proper dimensionality to retain. Seventy percent is often mentioned as the amount of variance in the data that should be used to select the optimal dimensionality for recomputing the SVD.[38][39][40]

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