Foundations of Independent Component Analysis

2026-08-13

AI Summary

This study addresses the identifiability of source signals in linear independent component analysis (ICA). Building upon measure-theoretic probability and characteristic function theory, the authors systematically develop a hierarchy of identifiability conditions ranging from weak to strong assumptions. Under the strongest conditions, they establish that source signals can be uniquely recovered up to permutation, scaling, sign, and translation—even in the presence of additive Gaussian noise—and precisely delineate the critical roles of non-Gaussianity and the absence of Gaussian components. The work further introduces a novel online equivariant gradient descent algorithm that efficiently recovers the sources, thereby unifying and extending the existing theoretical framework of ICA.

Abstract

We present the mathematical foundations of linear independent component analysis (ICA) models based on standard literature in a self-contained note. It is aimed at readers with a background in measure-theoretic probability theory. We first develop the theory of the characteristic functions of probability measures on $\mathbb{R}^d$, including their analyticity and the way in which they determine and characterise the distributions. We then focus on several identifiability results of ICA models with successively strengthened assumptions on the sources: from merely non-constant, to non-Gaussian, to Gaussian-free independent sources. Under the strictest assumptions, we show that the independent sources are identifiable up to translation, permutation, scales and signs, and this even in the presence of additive Gaussian noise. Furthermore, we present the online equivariant gradient descent ICA algorithm for recovering the independent sources from data, in the standard complete noiseless non-Gaussian ICA setting.

Problem

• Independent Component Analysis
• Identifiability
• Non-Gaussian Sources
• Source Separation
• Probability Measures

Innovation

• Independent Component Analysis
• Identifiability
• Characteristic Functions
• Equivariant Gradient Descent
• Non-Gaussian Sources