Nonlinear and robust independent component analysis for stochastic processes
Keywords: latent variable model
independent component analysis
statistical robustness
inverse problem
functional data analysis
statistical independence
expected signature
blind source separation
nonlinear BSS
nonlinear ICA
uncertainty quantification

This thesis is organised into two main parts, which are both preceded by a joint introduction and a brief chapter on the signature analysis of time-ordered data.
In the first part, 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 ‘nondegenerate’ 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.
In the second part, we revisit the problem of blind source separation from the perspective of statistical robustness. Blind source separation (BSS) aims to recover an unobserved signal S from its mixture X = f (S) under the condition that the effecting transformation f is invertible but unknown. This being a basic problem with numerous practical applications, a fundamental issue is to understand how the solutions to this problem behave when their supporting statistical prior assumptions are violated. In the classical context of linear mixtures, we present a general framework to analyse such violations and quantify the effect they have on the blind recovery of S from X. Modelling S as a multidimensional stochastic process, we introduce an informative topology on the space of possible causes underlying a mixture X and show that the behaviour of a generic BSS-solution in response to general deviations from its defining structural assumptions can be profitably analysed in the form of explicit continuity guarantees with respect to this topology. This enables a flexible and convenient quantification of general model uncertainty scenarios and amounts to the first comprehensive robustness framework for BSS. Our theory is entirely constructive, and we demonstrate its utility with a number of statistical applications.