Using state space differential geometry for nonlinear blind source separation

Given a time series of multicomponent data, the usual objective of nonlinear blind source separation (BSS) is 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 data'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. If the curvature tensor is zero, there is only one possible set of source variables (up to permutations and transformations of individual components), and it is possible to compute these explicitly and to determine if they do have a factorizable phase space density function. The method is illustrated by using it to separate two simultaneous synthetic "utterances" recorded with a single microphone. A more general method that performs nonlinear multidimensional BSS is described in Appendix A, where it is illustrated with analytic and numerical examples. (c) 2008 American Institute of Physics.