A Theoretical Framework for Overcomplete Geometric BMMR

F.J. Theis and E.W. Lang, M.A. Lautenschlager (Germany), and C.G. Puntonet (Spain)


geometric independent component analysis, blind source separation, overcomplete independent component analysis


Geometric algorithms for linear quadratic independent component analysis (ICA) have recently received some at tention due to their pictorial description and their relative ease of implementation. The geometric approach to ICA has been proposed first by Puntonet and Prieto [11] in or der to separate linear mixtures. Recently it has been gener alized to overcomplete cases (overcomplete geoICA) with more sources than sensors [14]. Here, we put this algo rithm in the two-step framework from [13]. We gener alize the geometric theory of quadratic case from [12] to the overcomplete case showing that fixpoints of geomet ric ICA fulfill a so called geometric convergence condi tion, which the mixed images of the unit vectors satisfy, too. This leads to a conjecture claiming that in the super gaussian unimodal symmetric case there is only one stable fixpoint, thus demonstrating uniqueness of overcomplete geoICA after convergence.

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