Generic uniqueness of a structured matrix factorization and applications in blind source separation

TL;DR

This paper establishes conditions for the generic uniqueness of structured matrix factorizations, with applications in blind source separation and independent component analysis, using algebraic geometry tools.

Contribution

It introduces a set of sufficient conditions for generic uniqueness of structured matrix factorizations, applicable to blind source separation and joint matrix diagonalization.

Findings

Provides a relaxed generic uniqueness condition for joint matrix diagonalization.

Establishes generic uniqueness conditions for deterministic blind source separation methods.

Connects algebraic geometry concepts to signal processing problems.

Abstract

Algebraic geometry, although little explored in signal processing, provides tools that are very convenient for investigating generic properties in a wide range of applications. Generic properties are properties that hold "almost everywhere". We present a set of conditions that are sufficient for demonstrating the generic uniqueness of a certain structured matrix factorization. This set of conditions may be used as a checklist for generic uniqueness in different settings. We discuss two particular applications in detail. We provide a relaxed generic uniqueness condition for joint matrix diagonalization that is relevant for independent component analysis in the underdetermined case. We present generic uniqueness conditions for a recently proposed class of deterministic blind source separation methods that rely on mild source models. For the interested reader we provide some intuition on how the results are connected to their algebraic geometric roots.