ON THE UNIQUENESS OF THE MINIMUM OF THE INFORMATION-THEORETIC COST FUNCTION FOR THE SEPARATION OF MIXTURES OF NEARLY GAUSSIAN SIGNALS
Riccardo Boscolo, Vwani Roychowdhury · 2003
A large number of Independent Component Analysis (ICA) algorithms are based on the minimization of the statistical mutual information between the reconstructed signals, in order to achieve the source separation. While it has been demonstrated that a global minimum of such cost function will result in the separation of the statistically independent sources, it is an open problem to show that such cost function has a unique minimum (up to scaling and permutations of the signals). Without such result, there is no guarantee that the related ICA algorithms will not get stuck in local minima, and hence, return signals that are statistically dependent. We derive a novel result showing that for the special case of mixtures of two independent and identically distributed (i.i.d.) signals with symmetric, nearly gaussian probability density functions, such objective function has no local minima. This result is shown to yield a useful extension of the well-known entropy power inequality. 1.