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

2026-01-29

AI Summary

This work investigates the quantification of non-Gaussianity of empirical distributions in Wasserstein space and the identification of their nearest Gaussian approximations. Leveraging optimal transport theory and the conical geometry of the translation-invariant quadratic Wasserstein space, the problem is recast as an orthogonal projection onto the Gaussian cone. To this end, two novel geometric quantities—the relative Wasserstein angle and the projection distance—are introduced, revealing that conventional moment-matching Gaussians are not the closest in the W₂ sense. The authors further prove that any filled cone spanned by two rays in this space forms a flat manifold, enabling a rigorous definition of angles and inner products. A stochastic manifold optimization algorithm is developed via the semi-discrete dual formulation, combining closed-form solutions in one dimension with numerical optimization in higher dimensions. Experiments on synthetic and real data demonstrate that the resulting Gaussians outperform moment-matching counterparts in Fréchet Inception Distance, and that the relative Wasserstein angle exhibits greater robustness than the original Wasserstein distance.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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 Fr\'echet Inception Distance (FID) scores.

Problem

• Wasserstein distance
• Gaussian approximation
• non-Gaussianity
• optimal transport
• W_2-nearest Gaussian

Innovation

• Relative Wasserstein angle
• Gaussian approximation
• Optimal transport
• Manifold optimization
• Wasserstein geometry