A Theoretical Framework for Overcomplete Geometric BMMR
Fabian Joachim Theis, Elmar Wolfgang Lang, M. Lautenschlager, Carlos G. Puntonet · University of Regensburg Publication Server (University of Regensburg) · 2002
Geometric algorithms for linear quadratic independent component analysis (ICA) have recently received some attention due to their pictorial description and their relative ease of implementation. The geometric approach to ICA has been proposed first by Puntonet and Prieto [15] [17] in order to separate linear mixtures. Recently it has been generalized to overcomplete cases (overcomplete geoICA) with more sources than sensors [21]. Here, we put this algorithm in the two-step framework from [20]. We generalize the geometric theory of quadratic case from [19] to the overcomplete case showing that fixpoints of geometric ICA fulfill a so called geometric convergence condition, which the mixed images of the unit vectors satisfy, too. This leads to a conjecture claiming that in the supergaussian unimodal symmetric case there is only one stable fixpoint, thus demonstrating uniqueness of overcomplete geoICA after convergence.