Convergence of gradient-based block coordinate descent algorithms for non-orthogonal joint approximate diagonalization of matrices

TL;DR

This paper introduces gradient-based block coordinate descent algorithms for joint approximate diagonalization of matrices on complex Stiefel and special linear groups, proving their global convergence.

Contribution

It develops new BCD-G algorithms with convergence guarantees for joint diagonalization on complex manifolds, extending existing methods with a Riemannian gradient approach.

Findings

Established global and weak convergence of the proposed algorithms.

Designed three classes of BCD-G algorithms: BCD-GLU, BCD-GQU, BCD-GU.

Applicable to both complex and real matrix cases.

Abstract

In this paper, we propose a gradient-based block coordinate descent (BCD-G) framework to solve the joint approximate diagonalization of matrices defined on the product of the complex Stiefel manifold and the special linear group. Instead of the cyclic fashion, we choose a block optimization based on the Riemannian gradient. To update the first block variable in the complex Stiefel manifold, we use the well-known line search descent method. To update the second block variable in the special linear group, based on four kinds of different elementary transformations, we construct three classes: GLU, GQU and GU, and then get three BCD-G algorithms: BCD-GLU, BCD-GQU and BCD-GU. We establish the global and weak convergence of these three algorithms using the \L{}ojasiewicz gradient inequality under the assumption that the iterates are bounded. We also propose a gradient-based Jacobi-type framework to solve the joint approximate diagonalization of matrices defined on the special linear group. As in the BCD-G case, using the GLU and GQU classes of elementary transformations, we focus on the Jacobi-GLU and Jacobi-GQU algorithms and establish their global and weak convergence. All the algorithms and convergence results described in this paper also apply to the real case.