Identifiability for Blind Source Separation of Multiple Finite Alphabet Linear Mixtures

TL;DR

This paper provides a comprehensive combinatorial characterization of when blind source separation is identifiable in linear mixtures of finite alphabet sources, including unknown weights and sources, with practical criteria and probabilistic analysis.

Contribution

It offers the first complete characterization of identifiability conditions for finite alphabet linear mixtures with unknown parameters, including an explicit construction and probabilistic insights.

Findings

Identifiability conditions depend on the distribution of a hitting time.

Probability of identifiability converges exponentially fast to one.

Simulation results support theoretical findings.

Abstract

We give under weak assumptions a complete combinatorial characterization of identifiability for linear mixtures of finite alphabet sources, with unknown mixing weights and unknown source signals, but known alphabet. This is based on a detailed treatment of the case of a single linear mixture. Notably, our identifiability analysis applies also to the case of unknown number of sources. We provide sufficient and necessary conditions for identifiability and give a simple sufficient criterion together with an explicit construction to determine the weights and the source signals for deterministic data by taking advantage of the hierarchical structure within the possible mixture values. We show that the probability of identifiability is related to the distribution of a hitting time and converges exponentially fast to one when the underlying sources come from a discrete Markov process. Finally, we explore our theoretical results in a simulation study. Our work extends and clarifies the scope of scenarios for which blind source separation becomes meaningful.