Contrast-Free ICA and Causal Inference via Wasserstein Distances to the Gaussian

Abstract

We study the squared $2$-Wasserstein distance to the standard Gaussian as a non-Gaussianity criterion and use it for linear Independent Component Analysis (ICA) and causal inference in Linear Non-Gaussian Acyclic Models (LiNGAM). The analysis relies on a strict inequality between the Wasserstein non-Gaussianity of independent standardized sources and that of their linear combinations. When at most one source is Gaussian, any unit-norm linear combination involving at least two sources has strictly smaller squared Wasserstein distance than the corresponding weighted sum of source distances. At the population level, this yields exact identification of the ICA unmixing matrix, up to signed permutation, and gives an analogous characterization of causal orders through least-squares residuals. We then define empirical plug-in estimators and prove distribution-free uniform convergence bounds under finite-moment assumptions, before detailing three practical solvers: a Picard-style orthogonal optimizer for ICA, an exhaustive dynamic program for causal-order search, and a greedy order-search variant. Empirically, we demonstrate competitive performance for both tasks and provide open-source implementations for source separation and causal inference.