How to generalize Geometric ICA to higher dimensions
Fabian Joachim Theis, Elmar Wolfgang Lang · University of Regensburg Publication Server (University of Regensburg) · 2002
Geometric algorithms for linear 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 [6] in order to separate linear mixtures. One major drawback of geometric algorithms is, however, an exponentially rising number of samples and convergence times with increasing dimensiononality thus basically restricting geometric ICA to low-dimensional cases. We propose to apply overcomplete ICA to geometric ICA [7] to reduce high-dimensional problems to lower-dimensional ones, thus generalizing geometric ICA to higher dimensions. 1 Basics For m,n # N let Mat(mn) be the R-vectorspace of real mn matrices, and Gl(n) := {W # Mat(n n) | det(W ) #= 0} be the general linear group of R n . In linear blind source separation (BSS), a random vector X : # # R m (mixed vector) originates from an independent random vector S : # # R n (source vector) by mixing with a mixing matrix A # Mat(m n), i.e.