Diagonality measures of Hermitian positive-definite matrices with application to the approximate joint diagonalization problem

Khaled Alyani (1) , Marco Congedo (2) , Maher Moakher (1)

1 LR-LAMSIN-ENIT - Laboratoire de Modélisation Mathématique et Numérique dans les Sciences de l'Ingénieur [Tunis]
2 GIPSA-VIBS - GIPSA - Vision and Brain Signal Processing

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 α-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.

Mots clés: Diagonality measure; positive-definite matrices; Riemannian; distance; Bhattacharyya distance; Kullback-Leibler divergence; log-det; α-divergence; approximate joint diagonalization; 94A17

Domaines: Traitement du signal et de l'image [eess.SP]; Neurosciences

Khaled Alyani, Marco Congedo, Maher Moakher. Diagonality measures of Hermitian positive-definite matrices with application to the approximate joint diagonalization problem. Linear Algebra and its Applications, 2017, 528, pp.290 - 320. ⟨10.1016/j.laa.2016.08.031⟩. ⟨hal-01578462⟩