Latent semantic analysis - AbsoluteAstronomy.com

Latent semantic analysis (LSA) is a technique in natural language processing

Natural language processing

Natural language processing is a field of computer science and linguistics concerned with the interactions between computers and human languages; it began as a branch of artificial intelligence....

, in particular in vectorial semantics, of analyzing relationships between a set of documents and the terms they contain by producing a set of concepts related to the documents and terms. LSA assumes that words that are close in meaning will occur close together in text. A matrix containing word counts per paragraph (rows represent unique words and columns represent each paragraph) is constructed from a large piece of text and a mathematical technique called singular value decomposition

Singular value decomposition

In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....

(SVD) is used to reduce the number of columns while preserving the similarity structure among rows. Words are then compared by taking the cosine of the angle between the two vectors formed by any two rows. Values close to 1 represent very similar words while values close to 0 represent very dissimilar words.

LSA was patented in 1988 (US Patent 4,839,853) by Scott Deerwester

Scott Deerwester

Scott Deerwester is one of the inventors of latent semantic analysis. He was a member of the faculty of the Colgate University, University of Chicago and the Hong Kong University of Science and Technology. He has been resident in Hong Kong since 1991, where he has been working in the humanitarian...

, Susan Dumais

Susan Dumais

Susan Dumais is a Principal Researcher in the Context, Learning, and User Experience for Search Group of Microsoft Research and an Affiliate Professor at the University of Washington Information School....

, George Furnas

George Furnas

Prof. George W. Furnas is a professor and Associate Dean for Academic Strategy at the School of Information of the University of Michigan. Prior to his position at the University of Michigan, Furnas worked at Bell Labs for 15 years where he was a distinguished member of technical staff, and then...

, Richard Harshman

Richard Harshman

Dr Richard A. Harshman was a member of the Department of Psychology of the University of Western Ontario since 1976, rising in the ranks to the level of Full Professor. He died suddenly on Thursday, January 10, 2008....

, Thomas Landauer

Thomas Landauer

Dr. Thomas K. Landauer is a professor at the Department of Psychology of the University of Colorado.He was one of the pioneers of Latent semantic analysis, and in 1995 published a controversial analysis of the productivity paradox of information technology.He is currently a Vice President at...

, Karen Lochbaum and Lynn Streeter. In the context of its application to information retrieval

Information retrieval

Information retrieval is the area of study concerned with searching for documents, for information within documents, and for metadata about documents, as well as that of searching structured storage, relational databases, and the World Wide Web...

, it is sometimes called Latent Semantic Indexing (LSI)

Latent semantic indexing

Latent Semantic Indexing is an indexing and retrieval method that uses a mathematical technique called Singular value decomposition 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...

Occurrence matrix

LSA can use a term-document matrix which describes the occurrences of terms in documents; it is a sparse matrix

Sparse matrix

In the subfield of numerical analysis, a sparse matrix is a matrix populated primarily with zeros . The term itself was coined by Harry M. Markowitz....

whose rows correspond to terms

Terminology

Terminology is the study of terms and their use. Terms are words and compound words that in specific contexts are given specific meanings, meanings that may deviate from the meaning the same words have in other contexts and in everyday language. The discipline Terminology studies among other...

and whose columns correspond to documents. A typical example of the weighting of the elements of the matrix is tf-idf (term frequency–inverse document frequency): the element of the matrix is proportional to the number of times the terms appear in each document, where rare terms are upweighted to reflect their relative importance.

This matrix is also common to standard semantic models, though it is not necessarily explicitly expressed as a matrix, since the mathematical properties of matrices are not always used.

Rank lowering

After the construction of the occurrence matrix, LSA finds a low-rank approximation to the term-document matrix. There could be various reasons for these approximations:

The original term-document matrix is presumed too large for the computing resources; in this case, the approximated low rank matrix is interpreted as an approximation (a "least and necessary evil").
The original term-document matrix is presumed noisy: for example, anecdotal instances of terms are to be eliminated. From this point of view, the approximated matrix is interpreted as a de-noisified matrix (a better matrix than the original).
The original term-document matrix is presumed overly sparse
Sparse matrix
In the subfield of numerical analysis, a sparse matrix is a matrix populated primarily with zeros . The term itself was coined by Harry M. Markowitz....

relative to the "true" term-document matrix. That is, the original matrix lists only the words actually in each document, whereas we might be interested in all words related to each document—generally a much larger set due to synonymy.

The consequence of the rank lowering is that some dimensions are combined and depend on more than one term:

{(car), (truck), (flower)} --> {(1.3452 * car + 0.2828 * truck), (flower)}

This mitigates the problem of identifying synonymy, as the rank lowering is expected to merge the dimensions associated with terms that have similar meanings. It also mitigates the problem with polysemy

Polysemy

Polysemy is the capacity for a sign or signs to have multiple meanings , i.e., a large semantic field.Charles Fillmore and Beryl Atkins’ definition stipulates three elements: the various senses of a polysemous word have a central origin, the links between these senses form a network, and ...

, since components of polysemous words that point in the "right" direction are added to the components of words that share a similar meaning. Conversely, components that point in other directions tend to either simply cancel out, or, at worst, to be smaller than components in the directions corresponding to the intended sense.

Derivation

Let

be a matrix where element

describes the occurrence of term

in document

(this can be, for example, the frequency).

will look like this:

Now a row in this matrix will be a vector corresponding to a term, giving its relation to each document:

Likewise, a column in this matrix will be a vector corresponding to a document, giving its relation to each term:

Now the dot product

Dot product

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

between two term vectors gives the correlation

Correlation

In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

between the terms over the documents. The matrix product

contains all these dot products. Element

(which is equal to element

) contains the dot product

(

). Likewise, the matrix

contains the dot products between all the document vectors, giving their correlation over the terms:

.

Now assume that there exists a decomposition of

such that

and

are orthogonal matrices

Orthogonal matrix

In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

and

is a diagonal matrix

Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...

. This is called a singular value decomposition

Singular value decomposition

In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....

(SVD):

The matrix products giving us the term and document correlations then become

Since

and

are diagonal we see that

must contain the eigenvectors of

, while

must be the eigenvectors of

. Both products have the same non-zero eigenvalues, given by the non-zero entries of

, or equally, by the non-zero entries of

. Now the decomposition looks like this:

The values

are called the singular values, and

and

the left and right singular vectors.
Notice the only part of

that contributes to

is the

row.
Let this row vector be called

.
Likewise, the only part of

that contributes to

is the

column,

.
These are not the eigenvectors, but depend on all the eigenvectors.

It turns out that when you select the

largest singular values, and their corresponding singular vectors from

and

, you get the rank

approximation to X with the smallest error (Frobenius norm). This approximation has a minimal error. But more importantly we can now treat the term and document vectors as a "concept space". The vector

then has

entries, each giving the occurrence of term

in one of the

concepts. Likewise, the vector

gives the relation between document

and each concept. We write this approximation as

You can now do the following:

See how related documents and are in the concept space by comparing the vectors and (typically by cosine similarity
Vector space model
Vector space model is an algebraic model for representing text documents as vectors of identifiers, such as, for example, index terms. It is used in information filtering, information retrieval, indexing and relevancy rankings...

). This gives you a clustering of the documents.
Comparing terms and by comparing the vectors and , giving you a clustering of the terms in the concept space.
Given a query, view this as a mini document, and compare it to your documents in the concept space.

To do the latter, you must first translate your query into the concept space. It is then intuitive that you must use the same transformation that you use on your documents:

This means that if you have a query vector

, you must do the translation

before you compare it with the document vectors in the concept space. You can do the same for pseudo term vectors:

Applications

The new concept space typically can be used to:

Compare the documents in the concept space (data clustering
Data clustering
Cluster analysis or clustering is the task of assigning a set of objects into groups so that the objects in the same cluster are more similar to each other than to those in other clusters....

, document classification
Document classification
Document classification or document categorization is a problem in both library science, information science and computer science. The task is to assign a document to one or more classes or categories. This may be done "manually" or algorithmically...

).
Find similar documents across languages, after analyzing a base set of translated documents (cross language retrieval).
Find relations between terms (synonymy and polysemy
Polysemy
Polysemy is the capacity for a sign or signs to have multiple meanings , i.e., a large semantic field.Charles Fillmore and Beryl Atkins’ definition stipulates three elements: the various senses of a polysemous word have a central origin, the links between these senses form a network, and ...

).
Given a query of terms, translate it into the concept space, and find matching documents (information retrieval
Information retrieval
Information retrieval is the area of study concerned with searching for documents, for information within documents, and for metadata about documents, as well as that of searching structured storage, relational databases, and the World Wide Web...

).
Find the best similarity between small groups of terms, in a semantic way (i.e. in a context of a knowledge corpus), as for example in multi choice questions MCQ
MCQ
MCQ may refer to* McQ, a 1974 crime drama* McQ Inc, a American defense company based in Pennsylvania* Mathematical Citation Quotient, a measure of the impact of a mathematics journal* Multiple choice question* McQuary limit, or McQ limit...

answering model.

Synonymy and polysemy are fundamental problems in natural language processing

Natural language processing

Natural language processing is a field of computer science and linguistics concerned with the interactions between computers and human languages; it began as a branch of artificial intelligence....

Synonymy is the phenomenon where different words describe the same idea. Thus, a query in a search engine may fail to retrieve a relevant document that does not contain the words which appeared in the query. For example, a search for "doctors" may not return a document containing the word "physicians", even though the words have the same meaning.
Polysemy is the phenomenon where the same word has multiple meanings. So a search may retrieve irrelevant documents containing the desired words in the wrong meaning. For example, a botanist and a computer scientist looking for the word "tree" probably desire different sets of documents.

Applications in human memory

The use of Latent Semantic Analysis has been prevalent in the study of human memory, especially in areas of free recall

Free recall

Free recall is a basic paradigm in the psychological study of memory. In this paradigm, participants study a list of items on each trial, and then are prompted to recall the items in any order...

and memory search. Howard and Kahana found a positive correlation between the semantic similarity of two words (as measured by LSA) and the probability that the words would be recalled one after another in free recall tasks using study lists of random common nouns. They also noted that in these situations, the inter-response time between the similar words was much quicker than between dissimilar words. These findings are referred to as the Semantic Proximity Effect.

Expanding on these semantic proximity effects, in 2006 Zaromb et al. found that when participants made mistakes in recalling studied items, these mistakes tended to be items that were more semantically related to the desired item and found in a previously studied list. These prior-list intrusions, as they have come to be called, seem to compete with items on the current list for recall. Zaromb et al. found that the PLIs had larger cosine values to the just-recalled word in LSA space than the correct item did.

Another model, termed Word Association Spaces (WAS) is also used in memory studies and was developed by Douglas Nelson at the University of South Florida by collecting free association data from a series of experiments and which includes measures of word relatedness for over 72,000 distinct word pairs.

Implementation

The SVD

Singular value decomposition

In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....

is typically computed using large matrix methods (for example, Lanczos methods) but may also be computed incrementally and with greatly reduced resources via a neural network

Neural network

The term neural network was traditionally used to refer to a network or circuit of biological neurons. The modern usage of the term often refers to artificial neural networks, which are composed of artificial neurons or nodes...

-like approach, which does not require the large, full-rank matrix to be held in memory.
A fast, incremental, low-memory, large-matrix SVD algorithm has recently been developed. MATLAB and Python implementations of these fast algorithms are available. Unlike Gorrell and Webb's (2005) stochastic approximation, Brand's algorithm (2003) provides an exact solution.

Limitations

Some of LSA's drawbacks include:

The resulting dimensions might be difficult to interpret. For instance, in

{(car), (truck), (flower)} --> {(1.3452 * car + 0.2828 * truck), (flower)}

the (1.3452 * car + 0.2828 * truck) component could be interpreted as "vehicle". However, it is very likely that cases close to

{(car), (bottle), (flower)} --> {(1.3452 * car + 0.2828 * bottle), (flower)}

will occur. This leads to results which can be justified on the mathematical level, but have no interpretable meaning in natural language.

LSA cannot capture polysemy
Polysemy
Polysemy is the capacity for a sign or signs to have multiple meanings , i.e., a large semantic field.Charles Fillmore and Beryl Atkins’ definition stipulates three elements: the various senses of a polysemous word have a central origin, the links between these senses form a network, and ...

(i.e., multiple meanings of a word). Each occurrence of a word is treated as having the same meaning due to the word being represented as a single point in space. For example, the occurrence of "chair" in a document containing "The Chair of the Board" and in a separate document containing "the chair maker" are considered the same. The behavior results in the vector representation being an average of all the word's different meanings in the corpus, which can make it difficult for comparison. However, the effect is often lessened due to words having a predominant sense
Word sense disambiguation
In computational linguistics, word-sense disambiguation is an open problem of natural language processing, which governs the process of identifying which sense of a word is used in a sentence, when the word has multiple meanings...

throughout a corpus (i.e. not all meanings are equally likely).

Limitations of bag of words model
Bag of words model
The bag-of-words model is a simplifying assumption used in natural language processing and information retrieval. In this model, a text is represented as an unordered collection of words, disregarding grammar and even word order.The bag-of-words model is used in some methods of document...

(BOW), where a text is represented as an unordered collection of words.

The probabilistic model of LSA does not match observed data: LSA assumes that words and documents form a joint Gaussian model (ergodic hypothesis
Ergodic hypothesis
In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a particle in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a...

), while a Poisson distribution
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...

has been observed. Thus, a newer alternative is probabilistic latent semantic analysis
Probabilistic latent semantic analysis
Probabilistic latent semantic analysis , also known as probabilistic latent semantic indexing is a statistical technique for the analysis of two-mode and co-occurrence data. PLSA evolved from latent semantic analysis, adding a sounder probabilistic model...

, based on a multinomial model, which is reported to give better results than standard LSA.

Commercial Applications

LSA has been used to assist in performing prior art

Prior art

Prior art , in most systems of patent law, constitutes all information that has been made available to the public in any form before a given date that might be relevant to a patent's claims of originality...

searches for patents.

Occurrence matrix

Rank lowering

Derivation

Applications

Applications in human memory

Implementation

Limitations

Commercial Applications

See also

Articles on LSA

Talks and Demonstrations

Implementations