Singular vector spaces for computing the structured distance to singularity

TL;DR

This paper introduces a novel framework using singular vector spaces to efficiently compute the structured distance to singularity in matrices, significantly improving speed while maintaining solution quality.

Contribution

It proposes a new approach based on singular vector spaces, providing theoretical insights and a practical, faster algorithm for structured distance to singularity.

Findings

The new algorithm is significantly faster than existing methods.

It maintains comparable accuracy in the computed structured distance.

Enables solving larger problems than previously possible.

Abstract

Finding the distance to singularity for a matrix is a ubiquitous problem in numerical linear algebra, and is elegantly solved by the Eckart-Young-Mirsky theorem. Its structured variant naturally emerges when one considers structured matrices, and wants to preserve their structure. Recent work has shown that this problem is particularly important for a class of matrix nearness problems that either entirely or partly reduce to a structured distance to singularity problem. In this work, we propose a new framework for addressing this problem, based on the concept of singular vector spaces, that is, linear subsets of the set of singular matrices. We analyze singular vector spaces in the context of this problem, prove new results, and detail how a specific subfamily of singular vector spaces can be incorporated into a practical algorithm. The resulting algorithm is based on globally minimizing a certain objective function alternatingly in its arguments. Numerical experiments demonstrate that this new algorithm is remarkably faster than the state-of-the-art, while the quality of the output remains comparable. This makes it possible to solve problems of much larger size than what was previously possible.