Local Convergence Properties of Fastica and Some Generalisations

In recent years, algorithms to perform Independent Component Analysis in blind identification, localisation of sources or more general in data analysis have been developed. Prominent example certainly is the socalled FastICA algorithms from the Finnish school. In this paper we will generalise the FastICA algorithm considered as a discrete dynamical system on the unit sphere to the case where all units converge simultaneously, i.e., we consider some kind of parallel FastICA algorithm living on the orthogonal group. In addition we present a local convergence analysis for the algorithms proposed in this paper building on earlier work. It turns out that one can treat these type of algorithms in a similar manner as the Rayleigh quotient iteration, well known in numerical linear algebra, i.e. considering the algorithm as a discrete dynamical system on a suitable manifold. The algorithms presented here are compared by several numerical experiments and simulations.