Nonlinear Independent Component Analysis for Discrete-Time and Continuous-Time Signals
This work provides theoretical guarantees and a new method for nonlinear Independent Component Analysis, which is a foundational problem in signal processing and machine learning for separating mixed signals.
This paper addresses the recovery of a multidimensional source signal from nonlinear mixtures, demonstrating that recovery is possible under conditions of sufficient differentiability and invertibility of the mixing function, along with specific 'non-degenerate' second-order statistics of the independent source components. The authors reformulate the problem as an optimization-based function approximation and propose a new objective function based on cumulant-like statistics to quantify mutual statistical dependence, leading to a scalable method for nonlinear Independent Component Analysis.
We study the classical problem of recovering a multidimensional source signal from observations of nonlinear mixtures of this signal. We show that this recovery is possible (up to a permutation and monotone scaling of the source's original component signals) if the mixture is due to a sufficiently differentiable and invertible but otherwise arbitrarily nonlinear function and the component signals of the source are statistically independent with 'non-degenerate' second-order statistics. The latter assumption requires the source signal to meet one of three regularity conditions which essentially ensure that the source is sufficiently far away from the non-recoverable extremes of being deterministic or constant in time. These assumptions, which cover many popular time series models and stochastic processes, allow us to reformulate the initial problem of nonlinear blind source separation as a simple-to-state problem of optimisation-based function approximation. We propose to solve this approximation problem by minimizing a novel type of objective function that efficiently quantifies the mutual statistical dependence between multiple stochastic processes via cumulant-like statistics. This yields a scalable and direct new method for nonlinear Independent Component Analysis with widely applicable theoretical guarantees and for which our experiments indicate good performance.