Heavy-tailed Independent Component Analysis

TL;DR

This paper introduces new algorithms for heavy-tailed independent component analysis that operate under weaker moment assumptions than previous methods, broadening the applicability of ICA in practical, heavy-tailed data scenarios.

Contribution

The paper presents the first provably efficient ICA algorithms that work with only $(1+ ext{gamma})$-moments or no moment assumptions at all, extending ICA's theoretical foundations.

Findings

Algorithms work under finite $(1+ ext{gamma})$-moment condition.

An algorithm that requires no moment assumptions when $A$ has orthogonal columns.

Techniques involve convex geometry and properties of multivariate Gaussian distributions.

Abstract

Independent component analysis (ICA) is the problem of efficiently recovering a matrix $A \in \mathbb{R}^{n\times n}$ from i.i.d. observations of $X=AS$ where $S \in \mathbb{R}^n$ is a random vector with mutually independent coordinates. This problem has been intensively studied, but all existing efficient algorithms with provable guarantees require that the coordinates $S_i$ have finite fourth moments. We consider the heavy-tailed ICA problem where we do not make this assumption, about the second moment. This problem also has received considerable attention in the applied literature. In the present work, we first give a provably efficient algorithm that works under the assumption that for constant $\gamma > 0$, each $S_i$ has finite $(1+\gamma)$-moment, thus substantially weakening the moment requirement condition for the ICA problem to be solvable. We then give an algorithm that works under the assumption that matrix $A$ has orthogonal columns but requires no moment assumptions. Our techniques draw ideas from convex geometry and exploit standard properties of the multivariate spherical Gaussian distribution in a novel way.