Connecting Kaporin's condition number and the Bregman log determinant divergence
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 minimize 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 minimizers of the aforementioned divergence, and we show that they are only a constant scaling from also minimizingKaporin's condition number. We highlight connections to information geometry and comment on future directions.
Fields of Study & Subjects
Statistical Mechanics and Entropy
Random Matrices and Applications
Stochastic Gradient Optimization Techniques
Publication details
- Journal / source
- Electronic Journal of Linear Algebra
- Publisher
- University of Wyoming Libraries
- Volume
- 42
- Pages
- 430–445
- Publication date
- 22 May 2026
- Language
- EN