Approximation of Hermitian Matrices by Positive Semidefinite Matrices using Modified Cholesky Decompositions

TL;DR

This paper introduces a fast, scalable algorithm for approximating Hermitian matrices with positive semidefinite matrices using modified Cholesky decompositions, allowing bound specifications and optimizing error and condition number.

Contribution

It presents a novel algorithm that preserves sparsity, allows bound constraints, and balances approximation error with condition number, outperforming existing methods.

Findings

Outperforms existing algorithms in approximation quality

Preserves sparsity pattern for large and sparse matrices

Provides a Cholesky decomposition of the approximation as a useful byproduct

Abstract

A new algorithm to approximate Hermitian matrices by positive semidefinite Hermitian matrices based on modified Cholesky decompositions is presented. In contrast to existing algorithms, this algorithm allows to specify bounds on the diagonal values of the approximation. It has no significant runtime and memory overhead compared to the computation of a classical Cholesky decomposition. Hence it is suitable for large matrices as well as sparse matrices since it preserves the sparsity pattern of the original matrix. The algorithm tries to minimize the approximation error in the Frobenius norm as well as the condition number of the approximation. Since these two objectives often contradict each other, it is possible to weight these two objectives by parameters of the algorithm. In numerical experiments, the algorithm outperforms existing algorithms regarding these two objectives. A Cholesky decomposition of the approximation is calculated as a byproduct. This is useful, for example, if a corresponding linear equation should be solved. A fully documented and extensively tested implementation is available. Numerical optimization and statistics are two fields of application in which the algorithm can be of particular interest.