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Connecting Kaporin's condition number and the Bregman log determinant divergence

Andreas A. Bock, Martin S. Andersen

TL;DR

This work investigates the relationship between the Bregman log determinant divergence $\mathcal{D}_{LD}(A,P)$ and Kaporin's condition number for preconditioning SPD systems. It proves that $\mathcal{D}_{LD}(A,P) \ge \ln K(AP^{-1})$, with equality under trace normalization, thereby linking divergence-based and Kaporin-based objectives for PCG convergence. The authors develop low-rank plus positive-definite preconditioners, introduce a spectral-trace scaling via $\alpha$, and show that the optimal scaling minimizes both $\mathcal{D}_{LD}$ and Kaporin's number, while revealing invariances under positive scaling. The work also situates these results within a broader information-geometric framework and outlines practical extensions, including randomized and FSAI-inspired approaches for scalable preconditioning.

Abstract

This paper presents some theoretical results relating the Bregman log determinant matrix divergence to Kaporin's condition number. These can be viewed as nearness measures between a preconditioner and a given matrix, and we show under which conditions these two functions coincide. We also give examples of constraint sets over which it is equivalent to minimise these two objectives. We focus on preconditioners that are the sum of a positive definite and low-rank matrix, which were developed in a previous work. These were constructed as minimisers of the aforementioned divergence, and we show that they are only a constant scaling from also minimising Kaporin's condition number. We highlight connections to information geometry and comment on future directions.

Connecting Kaporin's condition number and the Bregman log determinant divergence

TL;DR

This work investigates the relationship between the Bregman log determinant divergence and Kaporin's condition number for preconditioning SPD systems. It proves that , with equality under trace normalization, thereby linking divergence-based and Kaporin-based objectives for PCG convergence. The authors develop low-rank plus positive-definite preconditioners, introduce a spectral-trace scaling via , and show that the optimal scaling minimizes both and Kaporin's number, while revealing invariances under positive scaling. The work also situates these results within a broader information-geometric framework and outlines practical extensions, including randomized and FSAI-inspired approaches for scalable preconditioning.

Abstract

This paper presents some theoretical results relating the Bregman log determinant matrix divergence to Kaporin's condition number. These can be viewed as nearness measures between a preconditioner and a given matrix, and we show under which conditions these two functions coincide. We also give examples of constraint sets over which it is equivalent to minimise these two objectives. We focus on preconditioners that are the sum of a positive definite and low-rank matrix, which were developed in a previous work. These were constructed as minimisers of the aforementioned divergence, and we show that they are only a constant scaling from also minimising Kaporin's condition number. We highlight connections to information geometry and comment on future directions.

Paper Structure

This paper contains 11 sections, 9 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Spectral and other condition numbers
  3. Some convergence results
  4. Kaporin's condition number and the log determinant divergence
  5. The log determinant divergence and convergence of PCG
  6. Preconditioners based on low-rank approximations
  7. Bregman log determinant divergence-based preconditioners
  8. Minimisation of the Bregman divergence and Kaporin's condition number
  9. A general result on trace scaling
  10. Summary and future directions
  11. Proof of \ref{['cor:error']}

Key Result

Theorem 1

In addition, when we have

Figures (1)

  • Figure 1: Numerical illustration of how $\alpha$ influences the values $\kappa_2(P_\alpha^{-{\frac{1}{2}}} AP_\alpha^{-{\frac{1}{2}}})$, $\mathcal{D}_\textnormal{LD}(A, P_\alpha)$, $\ln(K(P_\alpha^{-1} A))$.

Theorems & Definitions (21)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • proof
  • Corollary 4
  • proof
  • ...and 11 more