Anderson Acceleration of the Alternating Projections Method for Computing the Nearest Correlation Matrix

Higham, Nicholas J. and Strabić, Nataša (2016) Anderson Acceleration of the Alternating Projections Method for Computing the Nearest Correlation Matrix. Numerical Algorithms, 72 (4). pp. 1021-1042. ISSN 1572-9265

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Abstract

In a wide range of applications it is required to compute the nearest correlation matrix in the Frobenius norm to a given symmetric but indefinite matrix. Of the available methods with guaranteed convergence to the unique solution of this problem the easiest to implement, and perhaps the most widely used, is the alternating projections method. However, the rate of convergence of this method is at best linear, and it can require a large number of iterations to converge to within a given tolerance. We show that Anderson acceleration, a technique for accelerating the convergence of fixed-point iterations, can be applied to the alternating projections method and that in practice it brings a significant reduction in both the number of iterations and the computation time. We also show that Anderson acceleration remains effective, and indeed can provide even greater improvements, when it is applied to the variants of the nearest correlation matrix problem in which specified elements are fixed or a lower bound is imposed on the smallest eigenvalue. Alternating projections is a general method for finding a point in the intersection of several sets and ours appears to be the first demonstration that this class of methods can benefit from Anderson acceleration.

Item Type:	Article
Uncontrolled Keywords:	nearest correlation matrix, indefinite matrix, positive semidefinite matrix, Anderson acceleration, alternating projections method, Dykstra's correction
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Nick Higham
Date Deposited:	28 Jul 2016
Last Modified:	20 Oct 2017 14:13
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2490

Available Versions of this Item

• Anderson Acceleration of the Alternating Projections Method for Computing the Nearest Correlation Matrix. (deposited 11 Jun 2015)
  • Anderson Acceleration of the Alternating Projections Method for Computing the Nearest Correlation Matrix. (deposited 04 Aug 2015)
    • Anderson Acceleration of the Alternating Projections Method for Computing the Nearest Correlation Matrix. (deposited 09 Jan 2016)
      • Anderson Acceleration of the Alternating Projections Method for Computing the Nearest Correlation Matrix. (deposited 28 Jul 2016) [Currently Displayed]

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