Maximum Entropy and Minimal Mutual Information in a Nonlinear Model

In blind source separation, two different separation techniques are mainly used: Minimal Mutual Information (MMI), where minimization of the mutual output information yields an independent random vector, and Maximum Entropy (ME), where the output entropy is maximized. However, it is yet unclear why ME should solve the separation problem, ie. result in an independent vector. Amari has given a partial confirmation for ME in the linear case in [1], where he proves that under the assumption of vanishing expectancy of the sources ME does not change the solutions of MMI up to scaling and permutation. In this paper, we generalize Amari’s approach to nonlinear ICA problems, where random vectors have been mixed by output functions of layered neural networks. We show that certain solution points of MMI are kept fixed by ME if no scaling of the weight vectors is allowed. In general, ME however might leave those MMI solutions using diagonal weights in the first network layer. Therefore, we conclude this paper by suggesting that in nonlinear ME algorithms diagonal weights should be fixed in later epochs.