April 2023 Nonlinear independent component analysis for discrete-time and continuous-time signals
Alexander Schell, Harald Oberhauser
Author Affiliations +
Ann. Statist. 51(2): 487-518 (April 2023). DOI: 10.1214/23-AOS2256

Abstract

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 nonrecoverable 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.

Funding Statement

AS was financially supported by an Oxford-Cocker Graduate Scholarship and a Mathematical Institute Scholarship. HO is supported by the DataSig Program [EP/S026347/1] and the CIMDA collaboration between the City University of Hong Kong and the University of Oxford.

Acknowledgments

The authors would like to extend their gratitude to the Associate Editor, four anonymous referees, the Editor, and Aapo Hyvärinen for their very helpful comments and suggestions which helped to significantly improve the original version of this paper and its presentation.

Citation

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Alexander Schell. Harald Oberhauser. "Nonlinear independent component analysis for discrete-time and continuous-time signals." Ann. Statist. 51 (2) 487 - 518, April 2023. https://doi.org/10.1214/23-AOS2256

Information

Received: 1 June 2021; Revised: 1 December 2022; Published: April 2023
First available in Project Euclid: 13 June 2023

zbMATH: 07714169
MathSciNet: MR4600990
Digital Object Identifier: 10.1214/23-AOS2256

Subjects:
Primary: 62H25 , 62M99
Secondary: 60L10 , 62H05 , 62M45 , 62R10

Keywords: blind source separation , Functional data analysis , Independent component analysis , inverse problem , latent variable model , nonlinear BSS , nonlinear ICA , statistical independence , statistical inference for stochastic processes , unsupervised learning

Rights: Copyright © 2023 Institute of Mathematical Statistics

Vol.51 • No. 2 • April 2023
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