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

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.

Causal discovery under mean independence and linearity

Causal discovery methods such as LiNGAM identify causal structure from observational data by assuming mutually independent disturbances. This assumption is fragile: shared volatility, common scale effects, or other forms of dependence can cause the methods to recover the wrong causal order, even with infinite data. We introduce the Linear Mean-Independent Acyclic Model (LiMIAM), which replaces full independence with weaker one-sided mean-independence restrictions on the disturbances. Under finite-order consequences of these restrictions, source nodes are generically identifiable, and hence a compatible causal order can be recovered recursively. Our proof is constructive and leads to DirectLiMIAM, a sequential residual-based algorithm for causal discovery under dependent noise. In simulations with mean-independent but dependent disturbances, DirectLiMIAM outperforms LiNGAM methods. A large-scale empirical application to the oil market highlights the implausibility of the independence assumption and the ability of DirectLiMIAM to recover a realistic causal ordering, from policy to production and from prices to inflation.