Importance Sampling for a Monte Carlo Matrix Multiplication Algorithm, with Application to Information Retrieval

We perform importance sampling for a randomized matrix multiplication algorithm by Drineas, Kannan, and Mahoney and derive probabilities that minimize the expected value (with regard to the distributions of the matrix elements) of the variance. We compare these optimized probabilities with uniform p...

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Published in:	SIAM journal on scientific computing Vol. 33; no. 4; pp. 1689 - 1706
Main Authors:	Eriksson-Bique, Sylvester, Solbrig, Mary, Stefanelli, Michael, Warkentin, Sarah, Abbey, Ralph, Ipsen, Ilse C. F.
Format:	Journal Article
Language:	English
Published:	Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
Subjects:	Accuracy Algorithms Approximation Documents Exact sciences and technology Expected values Experiments Information retrieval Mathematics Methods of scientific computing (including symbolic computation, algebraic computation) Multiplication & division Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Numerical methods in probability and statistics Probability Random variables Sciences and techniques of general use 68W20 Monte Carlo method 62P30 Multiplication Numerical linear algebra query term-by-document matrix Information retrieval 15B48 Probability distribution randomized algorithm Stochastic method Algorithm Variance matching Numerical analysis Scientific computation 15A99 Distribution function Matrix method Sampling expected value 68Q87
ISSN:	1064-8275, 1095-7197
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Abstract	We perform importance sampling for a randomized matrix multiplication algorithm by Drineas, Kannan, and Mahoney and derive probabilities that minimize the expected value (with regard to the distributions of the matrix elements) of the variance. We compare these optimized probabilities with uniform probabilities and derive conditions under which the actual variance of the optimized probabilities is lower. Numerical experiments with query matching in information retrieval applications illustrate that the optimized probabilities produce more accurate matchings than the uniform probabilities and that they can also be computed efficiently.
AbstractList	We perform importance sampling for a randomized matrix multiplication algorithm by Drineas, Kannan, and Mahoney and derive probabilities that minimize the expected value (with regard to the distributions of the matrix elements) of the variance. We compare these optimized probabilities with uniform probabilities and derive conditions under which the actual variance of the optimized probabilities is lower. Numerical experiments with query matching in information retrieval applications illustrate that the optimized probabilities produce more accurate matchings than the uniform probabilities and that they can also be computed efficiently.
Author	Solbrig, Mary Ipsen, Ilse C. F. Eriksson-Bique, Sylvester Stefanelli, Michael Warkentin, Sarah Abbey, Ralph
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SubjectTerms	Accuracy Algorithms Approximation Documents Exact sciences and technology Expected values Experiments Information retrieval Mathematics Methods of scientific computing (including symbolic computation, algebraic computation) Multiplication & division Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Numerical methods in probability and statistics Probability Random variables Sciences and techniques of general use
Title	Importance Sampling for a Monte Carlo Matrix Multiplication Algorithm, with Application to Information Retrieval
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