Joint Independent Subspace Analysis: Uniqueness and Identifiability

Abstract

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

Schur decomposition of several matrices

“…. , n} coming form the model for statistical signal processing, called Joint Independent Subspace Analysis (JISA), see [11] for more details on the model and construction of such matrices. JISA model allows us to transform these cross-correlation matrices as follows:…”

Schur's Lemma for Coupled Reducibility and Coupled Normality

“…The multiset joint analysis framework, as opposed to the analysis of K distinct unrelated datasets, arises when a sufficient number of cross-correlations among datasets, i.e., A ij and B ij , i = j, are not zero. It turns out [6,9,10] that the identifiability and uniqueness of this model, in its simplest form, boil down to characterizing the set of solutions to the system of matrix equations (1), when the cross-correlations among the latent signals are subject to coupled (ir)reducibility. In other words, the coupled reducibility conditions that we introduce in this paper can be attributed with a physical meaning, and can be applied to real-world problems.…”

“…In other words, the coupled reducibility conditions that we introduce in this paper can be attributed with a physical meaning, and can be applied to real-world problems. We refer the reader to [6,10] for further details on the JISA model, and on the derivation of (1). In this paper, we consider scenarios more general than those required to address the signal processing problem in [6,10].…”

“…We refer the reader to [6,10] for further details on the JISA model, and on the derivation of (1). In this paper, we consider scenarios more general than those required to address the signal processing problem in [6,10]. The analysis in Section 6, which focuses on coupled normal matrices, is inspired by the JISA model in [6,10], in which the matrices A and B are Hermitian, and thus a special case of coupled normal.…”

“…In this paper, we consider scenarios more general than those required to address the signal processing problem in [6,10]. The analysis in Section 6, which focuses on coupled normal matrices, is inspired by the JISA model in [6,10], in which the matrices A and B are Hermitian, and thus a special case of coupled normal. A limited version of some of the results in this manuscript was presented orally in, e.g., [7,4,1] and in a technical report [6]; however, they were never published.…”