Perturbation Analysis for Matrix Joint Block Diagonalization

TL;DR

This paper develops a perturbation theory for the matrix joint block diagonalization problem (JBDP), providing error bounds, backward error analysis, and a condition number to assess the stability of approximate solutions in noisy settings.

Contribution

It introduces the first perturbation analysis framework for JBDP, including error bounds and condition number, based on a recent solution uniqueness condition.

Findings

Error bounds for approximate solutions are established.

Backward error analysis for JBDP is performed.

A condition number for JBDP is proposed.

Abstract

The matrix joint block diagonalization problem (JBDP) of a given matrix set $\mathcal{A}=\{A_i\}_{i=1}^m$ is about finding a nonsingular matrix $W$ such that all $W^{T} A_i W$ are block diagonal. It includes the matrix joint diagonalization problem (JBD) as a special case for which all $W^{T} A_i W$ are required diagonal. Generically, such a matrix $W$ may not exist, but there are practically applications such as multidimensional independent component analysis (MICA) for which it does exist under the ideal situation, i.e., no noise is presented. However, in practice noises do get in and, as a consequence, the matrix set is only approximately block diagonalizable, i.e., one can only make all $\widetilde{W}^{T} A_i\widetilde{W}$ nearly block diagonal at best, where $\widetilde{W}$ is an approximation to $W$, obtained usually by computation. This motivates us to develop a perturbation theory for JBDP to address, among others, the question: how accurate this $\widetilde{W}$ is. Previously such a theory for JDP has been discussed, but no effort has been attempted for JBDP yet. In this paper, with the help of a necessary and sufficient condition for solution uniqueness of JBDP recently developed in [Cai and Liu, {\em SIAM J. Matrix Anal. Appl.}, 38(1):50--71, 2017], we are able to establish an error bound, perform backward error analysis, and propose a condition number for JBDP. Numerical tests validate the theoretical results.