text: The matrix joint block diagonalization problem (JBDP) of a given matrix set @math is about finding a nonsingular matrix @math such that all @math are block diagonal. It includes the matrix joint diagonalization problem (JBD) as a special case for which all @math are required diagonal. Generically, such a matrix @math 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 @math nearly block diagonal at best, where @math is an approximation to @math , obtained usually by computation. This motivates us to develop a perturbation theory for JBDP to address, among others, the question: how accurate this @math 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, 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.
  @cite_15: 
  @cite_27: The matrix joint diagonalization is a fundamental notion for studying the blind signal processing and the independent component analysis. We reformulate some classical results on the topic in terms of shear matrices. These matrices appear naturally in differential geometry and group theory and the present paper deals with certain perturbations of joint diagonalizers for which the knowledge of shear matrices is crucial.
  @cite_2: We investigate the sensitivity of the problem of nonorthogonal (matrix) joint diagonalization (NOJD). First, we consider the uniqueness conditions for the problem of exact joint diagonalization (EJD), which is closely related to the issue of uniqueness in tensor decompositions. As a byproduct, we derive the well-known identifiability conditions for independent component analysis (ICA) based on an EJD formulation of ICA. We next introduce some known cost functions for NOJD and derive flows based on these cost functions for NOJD. Then we define and investigate the noise sensitivity of the stationary points of these flows. We show that the condition number of the joint diagonalizer and uniqueness of the joint diagonalizer as measured by modulus of uniqueness (as defined in this paper) affect the sensitivity. We also investigate the effect of the number of matrices on the sensitivity. Our numerical experiments confirm the theoretical results.While this paper was under review a few of its results were presented in ICASSP07 conference in Honolulu, HI .
  @cite_8: This paper proposes to generalize the notion of independent component analysis (ICA) to the notion of multidimensional independent component analysis (MICA). We start from the ICA or blind source separation (BSS) model and show that it can be uniquely identified provided it is properly parameterized in terms of one-dimensional subspaces. From this standpoint, the BSS ICA model is generalized to multidimensional components. We discuss how ICA standard algorithms can be adapted to MICA decomposition. The relevance of these ideas is illustrated by a MICA decomposition of ECG signals.
summary: Though tremendous efforts have gone to solve , their perturbation problems had received little or no attention in the past. In fact, today there are only a handful articles written on the perturbations of only. For , Cardoso @cite_8 presented a first order perturbation bound for a set of commuting matrices, and the result was later generalized by Russo @cite_27 . For general , using gradient flows, Afsari @cite_2 studied sensitivity via cost functions and obtained first order perturbation bounds for the diagonalizer. Shi and Cai @cite_15 investigated a normalized through a constrained optimization problem, and obtained an upper bound on certain distance between an approximate diagonalizer of a perturbed optimization problem and an exact diagonalizer of the unperturbed optimization problem.