Performing Nonlinear Blind Source Separation with Signal Invariants

TL;DR

This paper introduces a method for nonlinear blind source separation that uses local invariant functions derived from data correlations to identify and extract independent source signals from multicomponent time series.

Contribution

It presents a novel approach to nonlinear BSS based on local invariants and provides explicit construction of sources when constraints are satisfied.

Findings

Method successfully separates speech-like sounds from a single microphone

Constraints on local invariants characterize data separability

Explicit source construction from data is possible when constraints hold

Abstract

Given a time series of multicomponent measurements x(t), the usual objective of nonlinear blind source separation (BSS) is to find a "source" time series s(t), comprised of statistically independent combinations of the measured components. In this paper, the source time series is required to have a density function in (s,ds/dt)-space that is equal to the product of density functions of individual components. This formulation of the BSS problem has a solution that is unique, up to permutations and component-wise transformations. Separability is shown to impose constraints on certain locally invariant (scalar) functions of x, which are derived from local higher-order correlations of the data's velocity dx/dt. The data are separable if and only if they satisfy these constraints, and, if the constraints are satisfied, the sources can be explicitly constructed from the data. The method is illustrated by using it to separate two speech-like sounds recorded with a single microphone.