Relative Wasserstein Angle and the Problem of the $W_2$-Nearest Gaussian Distribution

We study the problem of quantifying how far an empirical distribution deviates from Gaussianity under the framework of optimal transport. By exploiting the cone geometry of the relative translation invariant quadratic Wasserstein space, we introduce two novel geometric quantities, the relative Wasserstein angle and the orthogonal projection distance, which provide meaningful measures of non-Gaussianity. We prove that the filling cone generated by any two rays in this space is flat, ensuring that angles, projections, and inner products are rigorously well-defined. This geometric viewpoint recasts Gaussian approximation as a projection problem onto the Gaussian cone and reveals that the commonly used moment-matching Gaussian can \emph{not} be the (W_2)-nearest Gaussian for a given empirical distribution. In one dimension, we derive closed-form expressions for the proposed quantities and extend them to several classical distribution families, including uniform, Laplace, and logistic distributions; while in high dimensions, we develop an efficient stochastic manifold optimization algorithm based on a semi-discrete dual formulation. Experiments on synthetic data and real-world feature distributions demonstrate that the relative Wasserstein angle is more robust than the Wasserstein distance and that the proposed nearest Gaussian provides a better approximation than moment matching in the evaluation of Fréchet Inception Distance (FID) scores.

💡 Research Summary

The paper introduces a novel geometric framework for quantifying how far an empirical probability distribution deviates from the Gaussian family, using tools from optimal transport (OT). Building on the translation‑invariant quadratic Wasserstein distance (W_2), the authors adopt the relative Wasserstein distance (RW_2) defined on the quotient space (P_2(\mathbb{R}^d)/!\sim_T) where all translates of a distribution are identified. In this quotient space every distribution (except the Dirac at the origin) lies on a ray emanating from the apex (