On the Identifiability of Nonlinear ICA: Sparsity and Beyond

TL;DR

This paper demonstrates that under certain structural sparsity assumptions, nonlinear ICA models can be identifiable up to permutation and component-wise transformations without auxiliary variables, supported by theoretical analysis and experiments.

Contribution

It introduces a novel identifiability result for nonlinear ICA based solely on mixing process assumptions like sparsity, without relying on auxiliary information.

Findings

Theoretical proof of identifiability under sparsity constraints.

Estimation methods validated through experiments.

Practical data examples suggest real-world applicability.

Abstract

Nonlinear independent component analysis (ICA) aims to recover the underlying independent latent sources from their observable nonlinear mixtures. How to make the nonlinear ICA model identifiable up to certain trivial indeterminacies is a long-standing problem in unsupervised learning. Recent breakthroughs reformulate the standard independence assumption of sources as conditional independence given some auxiliary variables (e.g., class labels and/or domain/time indexes) as weak supervision or inductive bias. However, nonlinear ICA with unconditional priors cannot benefit from such developments. We explore an alternative path and consider only assumptions on the mixing process, such as Structural Sparsity. We show that under specific instantiations of such constraints, the independent latent sources can be identified from their nonlinear mixtures up to a permutation and a component-wise transformation, thus achieving nontrivial identifiability of nonlinear ICA without auxiliary variables. We provide estimation methods and validate the theoretical results experimentally. The results on image data suggest that our conditions may hold in a number of practical data generating processes.