On Separation of Signal Sources Using Kernel Estimates of Probability Densities

Abstract

The discussion in this paper revolves around the notion of separation problems. The latter can be thought of as a unifying concept which includes a variety of important problems in applied mathematics. Thus, for example, the problems of classification, clustering, image segmentation, and discriminant analysis can all be regarded as separation problems in which one is looking for a decision boundary to be used in order to separate a set of data points into a number of (homogeneous) subsets described by different conditional densities. Since, in this case, the decision boundary can be defined as a hyperplane, the related separation problems can be regarded as geometric. On the other hand, the problems of source separation, deconvolution, and independent component analysis represent another subgroup of separation problems which address the task of separating algebraically mixed signals. The main idea behind the present development is to show conceptually and experimentally that both geometric and algebraic separation problems are very intimately related, since there exists a general variational approach based on which one can recover either geometrically or algebraically mixed sources, while only little needs to be modified to go from one setting to another.