Inversion of polynomial systems and separation of nonlinear mixtures of finite-alphabet sources

In this contribution, multiple-input multiple-output (MIMO) mixing systems are considered, which are instantaneous and nonlinear but polynomial. We first address the problem of invertibility, searching the inverse in the class of polynomial systems. It is shown that Gröbner bases techniques offer an attractive solution for testing the existence of an exact inverse and computing it. By noticing that any nonlinear mapping can be interpolated by a polynomial on a finite set, we tackle the general nonlinear case. Relying on a finite alphabet assumption of the input source signals, theoretical results on polynomials allow us to represent nonlinear systems as linear combinations of a finite set of monomials. We then generalize the first results to give a condition for the existence of an exact nonlinear inverse. The proposed method allows to compute this inverse in polynomial form. In the light of the previous results, we go further to the blind source separation problem. It is shown that for sources in a finite alphabet, the nonlinear problem is tightly connected with both problems of underdetermination and of dependent sources. We concentrate on the case of two binary sources, for which an easy solution can be found. By simulation, this solution is compared to techniques borrowed from classification methods.