An Algebraic Approach to Nonorthogonal General Joint Block Diagonalization

Abstract

The exact/approximate non-orthogonal general joint block diagonalization (nogjbd) problem of a given real matrix set A = {A i } m i=1 is to find a nonsingular matrix W ∈ R n×n (diagonalizer) such that W T A i W for i = 1, 2, . . . , m are all exactly/approximately block diagonal matrices with the same diagonal block structure and with as many diagonal blocks as possible. In this paper, we show that a solution to the exact/approximate nogjbd problem can be obtained by finding the exact/approximate solutions to the system of linear equations A i Z = Z T A i for i = 1, . . . , m, followed by a block diagonalization of Z via similarity transformation. A necessary and sufficient condition for the equivalence of the solutions to the exact nogjbd problem is established. Two numerical methods are proposed to solve the nogjbd problem, and numerical examples are presented to show the merits of the proposed methods.Key words. joint block diagonalization, tensor decomposition, independent component analysis AMS subject classifications. 15A21,15A69, 65F30.1. Introduction. The joint block diagonalization problem, also called the simultaneous block diagonalization problem, is a particular case of the block term decomposition (BTD) of a third order tensor [10,11,14,26]. Such problem has found many applications in independent subspace analysis (e.g., [4,13,30,31]) and semidefinite programming (e.g., [18,8,2,9]). To specify the problem, we name the problem by nine capital letters wwxyyzzzz. The first two letters, ww, indicate the type of the matrices in the matrix set, sy/he for real symmetric/complex Hermitian matrices, ge for general matrices. The second letter, x, indicates that the problem is solved in the exact sense or the approximate sense, e for the former and a for the latter. The next two letters, yy, indicate the type of the diagonalizer, no/nu for non-orthogonal/non-unitary matrix, o/u for orthogonal/unitary matrix(often left as blank). The last four letters, zzzz, indicate the computation performed, jd for joint diagonalization, jbd for joint block diagonalization, gjbd for general joint block diagonalization. Next, we first give some definitions, then formulate the wweyyjbd problem and the wweyygjbd problem mathematically.Definition 1.1. We call τ n = (n 1 , . . . , n t ) a partition of positive integer n if n 1 , n 2 , . . . , n t are all positive integers and the sum of them is n, i.e., t i=1 n i = n. The integer t is called the cardinality of the partition τ n , denoted by card(τ n ). The set of all partitions of n is denoted by T n .Definition 1.2. Given a partition τ n = (n 1 , . . . , n t ), for any matrix A of order n, define its block diagonal part and off-block-diagonal part associated with τ n as Bdiag τn (A) = diag(A 11 , . . . , A tt ), OffBdiag τn (A) = A − Bdiag τn (A), respectively, where A ii is of order n i for i = 1, . . . , t. A matrix A is referred to as a τ n -block diagonal matrix if OffBdiag τn (A) = 0.

Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial

“…For model , we define the following performance index to measure the quality of the computed diagonalizer W , which is used in the works of Cai and Liu and Cai et al: where V −1 = [ V 1 , V 2 ,…, V t ], W = [ W 1 , W 2 ,…, W t ], for i = 1,2,…, t , the vector ( π (1), π (2),…, π ( t )) is a permutation of {1,2,…, t } satisfying ( n π (1) , n π (2) ,…, n π ( t ) ) = τ n , and the expression subspace( E , F ) denotes the angle between two subspaces specified by the columns of E and F , which can be computed by the MATLAB function “subspace.”…”

“…All the numerical examples were carried out on a quad-core Intel Core i5-6300HQ running at 2.30 GHz with 3.87-GB RAM, using MATLAB R2014a with machine = 2.2 × 10 −16 . We compare the performance of PEAR (with and without refinement) with the second GJBD algorithm in the work of Cai and Liu, 21 namely, ⋆-commuting-based method with a conservative strategy, SCMC for short, and two algorithms for the JBD problem, namely, JBD-LM 40 and JBD-NCG. 4 For PEAR, three refinement loops are used to improve the quality of the diagonalizer.…”

“…To proceed, in what follows, we reuse some definitions and notations and reformulate the JBD problem and the GJBD problem mathematically as in the works of Cai and Liu and Cai et al…”

“…Even if the solutions returned by algebraic methods are of “low quality,” they are usually good initial guesses for optimization methods. In the current literature, the algebraic methods for the GJBD problem fall into two categories: One is based on matrix ∗‐algebra (see, for example, the works of de Klerk et al, Maehara and Murota, and Murota et al for the orthogonal GJBD problem and a recent generation by Cai and Liu for the nonorthogonal GJBD problem), and the other is based on a matrix polynomial (see the work of Cai et al for the Hermitian GJBD problem). In the former category, the null space of a linear operator needs to be computed, which requires flops; thus, for problems with large n values, such an approach will be quite expensive for computation.…”

“…To proceed, in what follows, we reuse some definitions and notations and reformulate the JBD problem and the GJBD problem mathematically as in the works of Cai and Liu 21 and Cai et al 22 Definition 1. We call n = (n 1 , n 2 , … , n t ) a partition of positive integer n if n 1 , n 2 , … , n t are all positive integers and their sum is n, that is, ∑ =1 = .…”