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Least-Squares Covariance Matrix Adjustment

Boyd, Stephen and Xiao, Lin (2005) Least-Squares Covariance Matrix Adjustment. SIAM Journal on Matrix Analysis and Applications, 27 (2). pp. 532-546. ISSN 0895-4798 http://resolver.caltech.edu/CaltechAUTHORS:BOYsiamjmaa05

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Abstract

We consider the problem of finding the smallest adjustment to a given symmetric $n \times n$ matrix, as measured by the Euclidean or Frobenius norm, so that it satisfies some given linear equalities and inequalities, and in addition is positive semidefinite. This least-squares covariance adjustment problem is a convex optimization problem, and can be efficiently solved using standard methods when the number of variables (i.e., entries in the matrix) is modest, say, under $1000$. Since the number of variables is $n(n+1)/2$, this corresponds to a limit around $n=45$. Malick [{\it SIAM J. Matrix Anal.\ Appl.,} 26 (2005), pp. 272--284] studies a closely related problem and calls it the semidefinite least-squares problem. In this paper we formulate a dual problem that has no matrix inequality or matrix variables, and a number of (scalar) variables equal to the number of equality and inequality constraints in the original least-squares covariance adjustment problem. This dual problem allows us to solve far larger least-squares covariance adjustment problems than would be possible using standard methods. Assuming a modest number of constraints, problems with $n=1000$ are readily solved by the dual method. The dual method coincides with the dual method proposed by Malick when there are no inequality constraints and can be obtained as an extension of his dual method when there are inequality constraints. Using the dual problem, we show that in many cases the optimal solution is a low rank update of the original matrix. When the original matrix has structure, such as sparsity, this observation allows us to solve very large least-squares covariance adjustment problems.


Item Type:Article
Additional Information:© 2005 Society for Industrial and Applied Mathematics ∗Received by the editors June 11, 2004; accepted for publication (in revised form) by L. Vanderberghe April 4, 2005; published electronically November 22, 2005. We are grateful to Andrew Ng for suggesting the problem to us and to two anonymous reviewers for very useful suggestions and comments.
Subject Keywords:matrix nearness problems, covariance matrix, least-squares, semidefinite least-squares
Record Number:CaltechAUTHORS:BOYsiamjmaa05
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:BOYsiamjmaa05
Alternative URL:http://dx.doi.org/10.1137/040609902
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1382
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:13 Jan 2006
Last Modified:26 Dec 2012 08:44

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