Perturbation analysis for matrix joint block diagonalization

Abstract

The matrix joint block-diagonalization problem (jbdp) of a given matrix set A={Ai}i=1m is about finding a nonsingular matrix W such that all WTAiW are block-diagonal. It includes the matrix joint diagonalization problem (jdp) as a special case for which all WTAiW are required diagonal. Generically, such a matrix W may not exist, but there are practical 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 W˜TAiW˜ nearly block-diagonal at best, where W˜ is an approximation to W, obtained usually by computation. The main goal of this paper is to develop a perturbation theory for jbdp to address, among others, the question: how accurate this W˜ is. Previously such a theory for jdp has been discussed, but no effort had been attempted for jbdp because, in large part, there is no quantitative way to describe solution uniqueness of jbdp until 2017 when Cai and Liu (2017) [9] successfully obtained a necessary and sufficient uniqueness condition. Based on the condition, in this article, we will establish a perturbation theory for jbdp. Our main contributions include an error bound for the approximate block-diagonalizer W˜ and a backward error analysis for jbdp. Numerical tests are presented to validate the theoretical results.