On the Identifiability of Sparse ICA without Assuming Non-Gaussianity

TL;DR

This paper develops a new identifiability theory for sparse ICA that works with Gaussian sources using second-order statistics and introduces less restrictive structural assumptions, supported by estimation methods and experiments.

Contribution

It introduces a novel, less restrictive structural variability assumption enabling Gaussian source identifiability without non-Gaussianity assumptions.

Findings

Theoretical proof of identifiability under new assumptions

Two estimation methods based on second-order statistics and sparsity

Experimental validation of the theory and methods

Abstract

Independent component analysis (ICA) is a fundamental statistical tool used to reveal hidden generative processes from observed data. However, traditional ICA approaches struggle with the rotational invariance inherent in Gaussian distributions, often necessitating the assumption of non-Gaussianity in the underlying sources. This may limit their applicability in broader contexts. To accommodate Gaussian sources, we develop an identifiability theory that relies on second-order statistics without imposing further preconditions on the distribution of sources, by introducing novel assumptions on the connective structure from sources to observed variables. Different from recent work that focuses on potentially restrictive connective structures, our proposed assumption of structural variability is both considerably less restrictive and provably necessary. Furthermore, we propose two estimation methods based on second-order statistics and sparsity constraint. Experimental results are provided to validate our identifiability theory and estimation methods.