A Generalization to Schur's Lemma with an Application to Joint Independent Subspace Analysis

Dana Lahat (1) , Christian Jutten (1)

1 GIPSA-VIBS - GIPSA - Vision and Brain Signal Processing

This paper has a threefold contribution. First, it introduces a generalization to Schur's lemma from 1905 on irreducible representations. Second, it provides a comprehensive uniqueness analysis of a recently-introduced source separation model. Third, it reinforces the link between signal processing and representation theory, a field of algebra that is more often associated with quantum mechanics than with signal processing. The source separation model that this paper relies on performs joint independent subspace analysis (JISA) using second order statistics. In previous work, we derived the Fisher information matrix (FIM) that corresponds to this model. The uniqueness analysis in this paper is based on analysing the FIM, and the derivation is based on our proposed generalization to Schur's lemma. We provide proof both to the new lemma and to the uniqueness conditions. From a different perspective, the generalization to Schur's lemma is inspired by a coupled matrix block diagonalization problem that arises from the JISA model. The results in this paper generalize previous results about identifiability of independent vector analysis (IVA) using second order statistics. The results in this paper complement previously-known results on the uniqueness of joint block diagonalization (JBD) and block term decompositions (BTD), as well as of their coupled counterparts.

Mots clés: Uniqueness; blind source separation; independent subspace analysis; independent vector analysis; data fusion; irreducible representations; coupled decompositions; joint block diagonalization

Domaines: Traitement du signal et de l'image [eess.SP]

Dana Lahat, Christian Jutten. A Generalization to Schur's Lemma with an Application to Joint Independent Subspace Analysis. 2015. ⟨hal-01247899v1⟩