Researches And Applications Of Matrices And Tensors Approximate Joint Diagonalization Algorithms

Blind source separation(BSS)is an important issue that has been widely discussed in many literatures.It affects such as astronomy,biomedicine,seismology,spectroscopy and digital communications.A very important and effective way to deal with BSS problems is the approximate joint diagonalization(AJD)of a set of matrices or higher-order tensors.Besides in the BSS,these AJD algorithms are also used in many other important fields,such as image processing and independent component analysis(ICA).This thesis mainly considers their application in the BSS.The traditional AJD algorithms mainly focus on symmetric matrices,Hermitian matrices or symmetric higher-order tensors,which can only deal with single-set data BSS problems.However,the increasing availability of multiset and multimodal signals has posed significant challenges for conventional BSS methods which are originally designed to analyze single-set data.Therefore,the joint BSS(JBSS)algorithms have attracted great interests,which can simultaneously recover the underlying multiple variables from multiple datasets,in recent years.The purpose of this thesis is to extend the traditional AJD problem to that of non-Hermitian matrices and non-symmetric higher-order tensors,and then introduce efficient AJD algorithms applied in the JBSS.This thesis discussed the relationship between the AJD problems including non-Hermitian matrices and non-symmetric higher-order tensors and JBSS problems,and illustrated the difference from the traditional AJD algorithms which focus on symmetric matrices,Hermitian matrices or symmetric higher-order tensors.The primary contributions included in this thesis are summarized blow:1.This thesis introduced a non-Hermitian orthogonal AJD algorithm which can also be called as the approximate joint singular value decomposition(AJSVD)algorithm(N-AJSVD).A new parameter structure of unitary rotation matrix is proposed.This parameter structure only depends on one unknown parameter.Using the complex derivative method and a reasonable approximation technique,we obtained the analytical solution of the unknown parameter.The algorithm can obtain the optimal left and right rotation matrix at the same time,however,the traditional AJSVD algorithm based on Givens rotation matrix can only obtain the left and right Givens rotation matrix by alternately optimizing and updating.Therefore,the introduced algorithm can accelerate the convergence speed and can ensure higher accuracy.In addition,we applied the algorithm to the two datasets JBSS problem after pre-whitening,and proved its effectiveness through numerical experiments.2.This thesis introduced a non-Hermitian non-orthogonal AJD algorithm(NNAJD-ALS).The algorithm is based on the gradient and an optimal rank-1 approximation approach to minimize a least squares cost function.We also illustrated the effectiveness of the algorithm for the canonical polyadic decomposition(CPD)of the third-order tensor,and compared it with the traditional CPD algorithm in numerical experiments.It can be seen that the introduced algorithm outperforms the traditional CPD algorithm in terms of stability and accuracy.In addition,we applied the algorithm to the problem of two datasets JBSS without pre-whitening.Compared with some existing classic JBSS algorithms,the overall performance of the introduced algorithm is more competitive.3.In this thesis,an orthogonal AJD algorithm(NOHTJD)for non-symmetric higher-order tensors is introduced.This algorithm,to some extent,can be regarded as an extension of the N-AJSVD algorithm on higher-order tensors.We described the relationship between the AJD of ( ? 3)order tensors and the CPD of + 1 order tensors with orthogonal factor matrices,and compared them with existing algorithms.In addition,we separate mixed source signals,based on the introduced algorithm,through jointly diagonalizing a set of time-delay cross-high-order cumulants established by observed signals(pre-whited)from multiple(? 3)datasets,directly.Taking four data sets as an example,the algorithm shows more competitive performance compared with the existing algorithms.