Diagonality Measures of Hermitian Positive-Definite Matrices with Application to the Approximate Joint Diagonalization Problem

TL;DR

This paper introduces invariant diagonality measures for Hermitian positive-definite matrices, providing closed-form expressions and applications to approximate joint diagonalization, with numerical methods and practical signal processing implications.

Contribution

It proposes new diagonality measures with invariance properties and applies them to formulate and solve the approximate joint diagonalization problem.

Findings

Closed-form expressions for diagonality measures.

Invariance properties of the proposed measures.

Numerical algorithms for joint diagonalization.

Abstract

In this paper, we introduce properly-invariant diagonality measures of Hermitian positive-definite matrices. These diagonality measures are defined as distances or divergences between a given positive-definite matrix and its diagonal part. We then give closed-form expressions of these diagonality measures and discuss their invariance properties. The diagonality measure based on the log-determinant $\alpha$-divergence is general enough as it includes a diagonality criterion used by the signal processing community as a special case. These diagonality measures are then used to formulate minimization problems for finding the approximate joint diagonalizer of a given set of Hermitian positive-definite matrices. Numerical computations based on a modified Newton method are presented and commented.