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

2026-07-14

AI Summary

This work proposes a linear causal inference method that dispenses with conventional contrast functions and unifies the tasks of source separation and causal ordering in Independent Component Analysis (ICA) and Linear Non-Gaussian Acyclic Models (LiNGAM). The key innovation lies in employing the squared 2-Wasserstein distance to the standard Gaussian distribution as a measure of non-Gaussianity, which—under the mild assumption that at most one source is Gaussian—rigorously guarantees model identifiability. The approach integrates Wasserstein distance theory, distribution-free uniform convergence bounds, and a Picard-type orthogonal optimizer, and introduces efficient dynamic programming and greedy algorithms for causal order search. Experiments demonstrate superior performance in both source separation and causal discovery, and the implementation is publicly available.

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.

Problem

• Independent Component Analysis
• Causal Inference
• LiNGAM
• Non-Gaussianity
• Wasserstein Distance

Innovation

• Wasserstein distance
• Independent Component Analysis
• Causal Inference
• Non-Gaussianity
• LiNGAM