Identifiability for Blind Source Separation of Multiple Finite Alphabet Linear Mixtures

Abstract

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.

A Bayesian framework for molecular strain identification from mixed diagnostic samples

“…If the noise n in (1) is zero (or small enough) and if the matrix M contains sufficiently many distinct rows, then in most cases the vector w can be easily solved by sorting the values in the measurement d and assigning them to w while discarding values that are binary combinations of already assigned values [16,17,20]. However, in our case, there is no guarantee that the matrix contains enough distinct rows for this approach to work.…”

Multiscale blind source separation

“…Before we introduce estimators for ω and f , we need to discuss identifiability of these parameters in the SBSSRmodel, that is, conditions when g determines them uniquely via g = m i=1 ω i f i . Although, deterministic finite alphabet instantaneous (linear) mixtures, i.e., σ = 0 in the SBSSR-model (4), received a lot of attention in the literature [66,53,72,21,42,36,58], a complete characterization of identifiability remained elusive and has been recently provided in [5], which will be briefly reviewed here as far as it is required for our purposes. Obviously, not every mixture g ∈ M in (2) is identifiable.…”

“…, f m (x)) ∈ A m are sufficiently well separated by the mixing weights ω. This is quantified by the alphabet separation boundary [5] ASB(ω) = ASB(ω, A) := min a =a ∈A m ω a − ω a .…”

“…A necessary identifiability condition in the SBSSR-model is ASB(ω) > 0 (see [5,Section 3.A]), where the size of ASB(ω) can be understood as a conditioning number for the difficulty of separating the sources in the SBSSRmodel, that is, the smaller ASB(ω), the more difficult separation of sources. Therefore, to quantify the estimation error of any method which serves the purposes in 1.…”

“…Then g = ω 1 f 1 + ω 2 f 2 = f 1 , independently of ω, and hence, ω cannot be determined from g. Therefore, a certain kind of variability of the sources f i is necessary to ensure identifiability of the mixing weights ω. We employ from [5] the following simple sufficient identifiability condition.…”