Adaptive unsupervised extraction of one component of a linear mixture with a single neuron

Extracting one specific component of a linear mixture is to isolate it due to the observation of several mixtures of all the components. This is done in an unsupervised way, based on the sole knowledge that the components are independent. The classical solution is independent component analysis which extracts the components all at the same time. In this paper, given at least as many sensors as components, we propose a simpler approach which independently extracts each component with one neuron. The weights of the neuron are optimized by minimizing an even polynomial of its output. The corresponding adaptive algorithm is an extended anti-Hebbian rule with very low complexity. It can extract any specific negative kurtosis component. Global stability of the algorithm is investigated as well as steady-state fluctuations. The influence of additive noise is also considered. These theoretical results are thoroughly confirmed by computer simulations.