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Singular vector spaces for computing the structured distance to singularity

Lauri Nyman

TL;DR

This work proposes a new framework for addressing the distance to singularity for a matrix problem, based on the concept of singular vector spaces, that is, linear subsets of the set of singular matrices.

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.

Singular vector spaces for computing the structured distance to singularity

TL;DR

This work proposes a new framework for addressing the distance to singularity for a matrix problem, based on the concept of singular vector spaces, that is, linear subsets of the set of singular matrices.

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.
Paper Structure (17 sections, 16 theorems, 49 equations, 3 figures, 1 table, 3 algorithms)

This paper contains 17 sections, 16 theorems, 49 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Linear structures inside the set of singular matrices
  3. Linear geometry in $\mathcal{S}_n(\mathbb{F})$
  4. Characterization of singular vector spaces
  5. Structured distance to singularity
  6. Problem statement
  7. New method
  8. Tikhonov regularization
  9. Augmented Lagrangian
  10. Convergence
  11. Numerical experiments
  12. Comparison of singular vector spaces
  13. Comparison of regularization approaches
  14. Comparison against existing methods
  15. Toeplitz structures
  16. ...and 2 more sections

Key Result

Proposition 2.1

A set $S \subset \mathbb{F}^{n}$ is a vector space if and only if for every pair of points $(x,y) \in S^2$ there exists an $\mathbb{F}$-affine subspace $A_{x,y}$ of $S$ such that $x,y \in A$.

Figures (3)

  • Figure 1: Comparison between the Tikhonov regularization approach and the augmented Lagrangian approach for Toeplitz structured matrices of increasing sizes. The median values of 40 runs were used for plotting.
  • Figure 2: Comparison between the singular vector space approach of this paper (denoted by SVS) and the Riemann-Oracle method of oracle (denoted by RO-L and RO-P) for Toeplitz structured matrices of increasing sizes. RO-L denotes the augmented Lagrangian formulation of oracle, while RO-P denotes its penalty method forumulation. The median values of 40 runs were used for plotting.
  • Figure 3: Comparison between the singular vector space approach of this paper (denoted by SVS), the Riemann-Oracle method of oracle (denoted by RO-L and RO-P) and the ODE approach of Sicilia for sparse matrices of increasing sizes. RO-L denotes the augmented Lagrangian formulation of oracle, while RO-P denotes its penalty method forumulation. The median values of 40 runs were used for plotting.

Theorems & Definitions (34)

  • Proposition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Theorem 2.5: flanders
  • Proposition 2.6: Fillmore
  • Proposition 2.7
  • Proof 1
  • Lemma 2.8
  • Proof 2
  • ...and 24 more