Using State Space Differential Geometry for Nonlinear Blind Source Separation

Abstract

Given a time series of multicomponent measurements of an evolving stimulus, nonlinear blind source separation (BSS) usually 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. In an earlier paper, it was shown that the phase space density function induces a Riemannian geometry on the system’s state space, with the metric equal to the local velocity correlation matrix of the data. From this geometric perspective, the vanishing of the curvature tensor is a necessary condition for BSS. 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 set of source variables (up to transformations that do not affect separability), and it is possible to compute these explicitly and determine if they do separate the phase space density function. A longer version of this paper describes a more general method that performs nonlinear multidimensional BSS or independent subspace separation.