Model-Independent Analytic Nonlinear Blind Source Separation

Abstract

Consider a time series of measurements of the state of an evolving system, x(t), where x has two or more components. This paper shows how to perform nonlinear blind source separation; i.e., how to determine if these signals are equal to linear or nonlinear mixtures of the state variables of two or more statistically independent subsystems. First, the local distributions of measurement velocities are processed in order to derive vectors at each point in x-space. If the data are separable, each of these vectors must be directed along a subspace of x-space that is traversed by varying the state variable of one subsystem, while all other subsystems are kept constant. Because of this property, these vectors can be used to construct a small set of mappings, which must contain the unmixing function, if it exists. Therefore, nonlinear blind source separation can be performed by examining the separability of the data after it has been transformed by each of these mappings. The method is analytic, constructive, and model-independent. It is illustrated by blindly recovering the separate utterances of two speakers from nonlinear combinations of their audio waveforms.