ICA 2007 

London, UK 9  12 September 2007 

Paper No: 15Using State Space Differential Geometry for Nonlinear Blind Source SeparationAuthor(s): David LevinAbstractGiven a time series of multicomponent measurements of an evolving stimulus, nonlinear blind source separation (BSS) seeks to find a "source" time series, comprised of statistically independent combinations of the measured components. In this paper, we seek a source time series that has a phasespace density function equal to the product of density functions of individual components. This criterion of statistical independence is stronger than that of conventional approaches to BSS, in which only the statespace density function is required to be separable. Because of the relative strength of this statistical criterion, the new approach to BSS produces a unique solution in each case (i.e., data are either inseparable or are separable by a unique mixing function), unlike the conventional approach that always leads to an infinite number of mixing functions. An earlier paper showed that a Riemannian geometry is induced on the state space by the local velocity correlation matrix, which can be taken to be the metric. From this geometric perspective, a necessary condition for BSS is the vanishing of the curvature tensor. Therefore, if this dataderived quantity is nonvanishing, the observations are not separable. However, if the curvature tensor is zero, there is only one possible separable coordinate system, and it is a geodesic coordinate system that can be constructed from the dataderived affine connection on state space. The data are separable if and only if the density function is seen to factorize in this geodesic coordinate system, in which case the geodesic coordinates are the unique source variables (up to transformations that do not affect separability: namely, translations, permutations, and rescaling of individual components). A longer version of this paper describes a more general method that performs nonlinear multidimensional BSS or independent subspace separation. 


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