From separability/identifiability properties of bilinear and linear-quadratic mixture matrix factorization to factorization algorithms

Abstract

Blind source separation (BSS) and Blind Mixture Identification (BMI) methods typically concern unknown source signals, transferred through a given class of functions with unknown parameter values, which yields mixed observations. Using only these observations, BSS/BMI aims at estimating the source signals and/or mixing parameters. Most investigations concern linear instantaneous mixing functions. They contain two aspects. The first one consists in proposing general BSS/BMI principles, e.g. Independent Component Analysis, Sparse Component Analysis or Nonnegative Matrix Factorization (NMF), and/or deriving associated practical algorithms. The second aspect consists in analyzing the properties resulting from these principles. This is of utmost importance, to determine if the proposed BSS/BMI principles are guaranteed to separate the source signals and to identify the considered mixing model up to acceptable indeterminacies. These separability/identifiability analyses are even more important for nonlinear mixtures, that were shown to potentially yield higher indeterminacies. Among them, bilinear and linear-quadratic mixtures are receiving increasing attention, e.g. due to their application to remote sensing. Especially, extensions of NMF were recently proposed for them, but the resulting separability/identifiability properties were not analyzed. We here address this topic, moreover proceeding further by investigating Bilinear and Linear-Quadratic Mixture Matrix Factorization (BMMF and LQMMF) approaches without nonnegativity constraints. We especially show that, whereas nonlinearity is often considered to be a burden, it yields an essentially unique decomposition under mild conditions for BMMF. On the contrary, full LQMMF is shown to yield spurious solutions, which increases the usefulness of combining it with nonnegativity constraints in applications where data meet these constraints. Algorithms based on this framework are also defined in this paper and their performance is reported.