Sparsity-Preserving Difference of Positive Semidefinite Matrix Representation of Indefinite Matrices

TL;DR

This paper introduces a novel difference-of-Cholesky method for representing indefinite symmetric matrices as the difference of two positive semidefinite matrices, enabling sparsity promotion and broad applicability.

Contribution

It develops a simple, adaptable difference-of-Cholesky algorithm for indefinite matrices, extending eigenvalue-based methods and allowing sparsity heuristics.

Findings

Effective difference-of-Cholesky representation for indefinite matrices

Sparsity-promoting heuristics can be integrated

Potential for improved matrix decomposition techniques

Abstract

We consider the problem of writing an arbitrary symmetric matrix as the difference of two positive semidefinite matrices. We start with simple ideas such as eigenvalue decomposition. Then, we develop a simple adaptation of the Cholesky that returns a difference-of-Cholesky representation of indefinite matrices. Heuristics that promote sparsity can be applied directly to this modification.