Nonlinear blind source separation for sparse sources

Abstract

Blind Source Separation (BSS) is the problem of separating signals which are mixed through an unknown function from a number of observations, without any information about the mixing model. Although it has been mathematically proven that the separation can be done when the mixture is linear, there is not any proof for the separability of nonlinearly mixed signals. Our contribution in this paper is performing nonlinear BSS for sparse sources. It is shown in this case, sources are separable even if the problem is under-determined (the number of observations is less that the number of source signals). However in the most general case (when the nonlinear mixing model can be of any kind and there is no side-information about that), an unknown nonlinear transformation of each source is reconstructed. It is shown why the problem reconstructing the exact sources is severely ill-posed and impossible to do without any other information.

Blind Source Separation in Nonlinear Mixtures: Separability and a Basic Algorithm

“…It should be mentioned that this work, as well as other general nonlinear BSS methods [2], [28], [29], [30], suffers from the ambiguity of a nonlinear transformation that cannot be resolved. However, it is important to differentiate between source separation and source reconstruction.…”

“…i,j ) can be calculated as a function of the parameters according to (29) and (30). Thus it can be solved, and the optimal parameter vectors ✓ ⇤ i,j will let us formulate the [J g;t (x)] i,j 's.…”

Blind compensation of polynomial mixtures of Gaussian signals with application in nonlinear blind source separation

“…While this problem has been well studied for linear mixtures and many algorithms have been already proposed in this regard [5,6,7,8,9,10], there is not much progress in general nonlinear cases. Consequently, except a few works on nonlinear BSS in general case [11,12], studies on this issue were focused on specific applications with restricted mixing models [13,14,15,16,17,18]. This is why it is thought that in general, roughly speaking, linear mixtures are separable while nonlinear ones are not.…”