Paper No: 15
Using State Space Differential Geometry for Nonlinear Blind Source Separation
Author(s): David Levin
Given 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 phase-space 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 state-space 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 data-derived quantity is non-vanishing, 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 data-derived 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.