Second Order Subspace Analysis and Simple Decompositions

Gutch, Harold W.; Maehara, Takanori; Theis, Fabian J.

doi:10.1007/978-3-642-15995-4_46

Cited by 6 publications

(14 citation statements)

References 9 publications

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“…We also use the vec{·} operator, which stacks the columns of a p × q matrix into a pq × 1 vector; the Kronecker product ⊗ and the property [11] vec{MXN} = (N † ⊗ M )vec{X} for any matrices M , N , X of compatible dimensions. It is assumed that H ij is invertible (which is usually the case for randomly-generated data; for further discussion see [12]). The set of matrices E ij , ∀i = j, obtained from (10), constitute the Newton step.…”

Section: S It Should Be Emphasized That Since Bdiag M {∇C(a)} Is Inmentioning

confidence: 99%

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Joint Block Diagonalization Algorithms for Optimal Separation of Multidimensional Components

Lahat

Jean-Fran$#

Cardoso

et al. 2012

Latent Variable Analysis and Signal Separation

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Abstract. This paper deals with non-orthogonal joint block diagonalization. Two algorithms which minimize the Kullback-Leibler divergence between a set of real positive-definite matrices and a block-diagonal transformation thereof are suggested. One algorithm is based on the relative gradient, and the other is based on a quasi-Newton method. These algorithms allow for the optimal, in the mean square error sense, blind separation of multidimensional Gaussian components. Simulations demonstrate the convergence properties of the suggested algorithms, as well as the dependence of the criterion on some of the model parameters.

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Section: S It Should Be Emphasized That Since Bdiag M {∇C(a)} Is Inmentioning

confidence: 99%

“…Therefore, for each trial we evaluate the normalized empirical and theoretical MSE. The normalized empirical MSE is defined as (12),…”

Section: Simulationsmentioning

confidence: 99%

Joint Block Diagonalization Algorithms for Optimal Separation of Multidimensional Components

Lahat

Jean-Fran$#

Cardoso

et al. 2012

Latent Variable Analysis and Signal Separation

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“…In this paper, we extend the concept of irreducibility to multiple transformations, and show that it is equally crucial in analysing JISA and CBD uniqueness and identifiability. In Section VIII, we discuss some of the links between the results in [19], [26], and [27] and those in this paper, and explain how the results in this paper can be regarded as a generalization of the former. CBD and JBD will be described more rigorously in the upcoming sections of this paper.…”

Section: Introductionmentioning

confidence: 87%

“…and Q is the number of distinct equations in (26). The factorization in (26) is often referred to as JBD (e.g., [25]). In analogy to (10), each summand in (25)…”

Section: A a Link Between Jisa And Isamentioning

confidence: 99%

“…In Section VIII-A, we show that these models are not only similar but in fact, in certain respects, ISA and JBD can be regarded as special cases of JISA and CBD. The uniqueness and identifiability results for these ISA and JBD models, in [19], [26], and [27], are based on Schur's lemma on irreducible representations [28], whose link with JBD was first pointed out in [29]. Schur's lemma deals with a single transformation applied to an ensemble of matrices, and is thus not applicable to CBD.…”

Section: Introductionmentioning

confidence: 99%

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Joint Independent Subspace Analysis: Uniqueness and Identifiability

Lahat¹,

Jutten

2019

IEEE Trans. Signal Process.

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This paper deals with the identifiability of joint independent subspace analysis (JISA). JISA is a recently-proposed framework that subsumes independent vector analysis (IVA) and independent subspace analysis (ISA). Each underlying mixture can be regarded as a dataset; therefore, JISA can be used for data fusion. In this paper, we assume that each dataset is an overdetermined mixture of several multivariate Gaussian processes, each of which has independent and identically distributed samples. This setup is not identifiable when each mixture is considered individually. Given these assumptions, JISA can be restated as coupled block diagonalization (CBD) of its correlation matrices. Hence, JISA identifiability is tantamount to CBD uniqueness. In this work, we provide necessary and sufficient conditions for uniqueness and identifiability of JISA and CBD. Our analysis is based on characterizing all the cases in which the Fisher information matrix is singular. We prove that non-identifiability may occur only due to pairs of underlying random processes with the same dimension. Our results provide further evidence that irreducibility has a central role in the uniqueness analysis of block-based decompositions. Our contribution extends previous results on the uniqueness and identifiability of ISA, IVA, coupled matrix and tensor decompositions. We provide examples to illustrate our results. Index Terms-Blind source separation, block decompositions, coupled decompositions, data fusion, identifiability, independent vector analysis, uniqueness.Dana Lahat received the Ph.D. degree in electrical engineering from Tel Aviv University, Tel Aviv, Israel, in 2013. She is currently a Postdoctoral Researcher with the Toulouse

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Perturbation analysis for matrix joint block diagonalization

Cai

2019

Linear Algebra and its Applications

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The matrix joint block diagonalization problem (jbdp) of a given matrix set A = {A i } m i=1 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 (jdp) 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, ie., 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 T A i W nearly block diagonal at best, where 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 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, 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.

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Second Order Subspace Analysis and Simple Decompositions

Cited by 6 publications

References 9 publications

Joint Block Diagonalization Algorithms for Optimal Separation of Multidimensional Components

Joint Block Diagonalization Algorithms for Optimal Separation of Multidimensional Components

Joint Independent Subspace Analysis: Uniqueness and Identifiability

Perturbation analysis for matrix joint block diagonalization

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