Non-Euclidean Independent Component Analysis and Oja's Learning

Abstract

In the present contribution we tackle the problem of nonlinear independent component analysis by non-Euclidean Hebbian-like learning. Independent component analysis (ICA) and blind source separation originally were introduced as tools for the linear unmixing of the signals to detect the underlying sources. Hebbian methods became very popular and succesfully in this context. Many nonlinear ICA extensions are known. A promising strategy is the application of kernel mapping. Kernel mapping realizes an usually nonlinear but implicite data mapping of the data into a reproducing kernel Hilbert space. After that a linear demixing can be carried out there. However, explicit handling in this non-Euclidean kernel mapping space is impossible. We show in this paper an alternative using an isomorphic mapping space. In particular, we show that the idea of Hebbian-like learning of kernel ICA can be transferred to this non-Euclidean space realizing an non-Euclidean ICA