Information Approach to Blind Source Separation and Deconvolution

Blind separation of sources aims to recover the sources from a their mixtures without relying on any specific knowledge of the sources and/or on the mixing mechanism [1]. (That is why the separation is called blind). Instead, it relies on the basic assumption that the sources are mutually independent.¹ A popular measure of dependence is the mutual information. This chapter attempts to provide a systematic approach to blind source separation based on the mutual information.

We shall focus on noiseless mixtures. There are few methods which deal explicitly with noises. Often a preprocessing step is done to reduce noises, or a method designed for a noiseless model is found to be rather insensitive to noise and can thus be applied when some noises are present. A general noiseless mixture model can be written as x(·) = A{s(·)}, where s(n) et x(n) represent the observation and the source vector at time n et A is some transformation, which can be instantaneous, i.e. operating on each s(n) to produce x(n), or global (i.e. operating on the whole sequence s(·) of the source vectors. The transformation A is not completely arbitrary, one often assumes it belongs to a certain class A, the most popular ones are the class of linear (or affine) instantaneous transformation and the class of linear convolutions. More complex non linear transformations have been considered, but for simplicity we shall limit ourselves to the above two linear classes. Separation may be realized by applying an inverse transformation A^-1 to x(·). However, A is unknown, so is its inverse. The natural idea is to apply a transformation B є A ^-1, the set of all transformations which are inverses of a transformation in A, and is chosen to minimize some criterion. We consider here the independence criterion based on the mutual information measure.