Heavy-tailed Independent Component Analysis
This addresses a limitation in ICA for handling heavy-tailed data, which is incremental but important for applications where finite fourth moments cannot be assumed.
The paper tackles the heavy-tailed independent component analysis (ICA) problem by developing efficient algorithms that work under weaker moment assumptions than existing methods, specifically requiring only finite (1+γ)-moment for each coordinate or no moment assumptions when the matrix has orthogonal columns.
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 $γ> 0$, each $S_i$ has finite $(1+γ)$-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.