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

TL;DR

This paper introduces a geometric framework for measuring non-Gaussianity in probability distributions under the 2-Wasserstein OT metric by defining the relative Wasserstein angle ($RW_2$ angle) and the orthogonal projection distance. By exploiting the cone structure of translations and dilations in the $RW_2$ space, the authors show that the filling cone between two rays is flat, enabling Euclidean-like notions of angles and projections and reframing Gaussian approximation as projection onto the Gaussian cone. They prove that the moment-matching Gaussian is typically not the $W_2$-nearest Gaussian, provide a closed-form solution in one dimension, and develop a dual-based, stochastic Riemannian optimization method for high dimensions. Experiments on synthetic data and real-world feature distributions demonstrate that the $W_2$-nearest Gaussian yields smaller $RW_2$ angles and projection distances than moment matching, offering more faithful Gaussian surrogates and improved robustness, especially for high-dimensional data. The framework has potential applications beyond Gaussian models, enabling robust, geometry-grounded distribution comparisons and enhancements to evaluation metrics like FID.

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 Fréchet Inception Distance (FID) scores.

Figures (4)

• Figure 1: Schematic illustration of the cone geometry in the $RW_2$ space. The apex $[\delta_0]$ represents the Dirac measure and the rays $[[\mu]]$ and $[[\nu]]$ emanate from this point. The shaded blue region between the two rays is the filling cone. The angle $\theta$ between the rays corresponds to the relative Wasserstein angle. The point $[\mu]$ lies on the ray $[[\nu]]$ at distance $\|[\mu]\|$ from the apex, and its orthogonal projection onto the ray $[[\mu]]$ has length $l = \|[\mu]\| \cos \theta$.
• Figure 2: Top: Evolution of the $RW_2$ angle $\theta$ and the projection distance $p$ as functions of the sample size $N$. Shaded regions indicate one standard deviation over repeated experiments. As $N$ increases, both quantities decrease, reflecting the decay of finite-sample non-Gaussianity. Bottom: Gaussian approximation at a fixed sample size $N = 64$. For each $m$, the solid curve represents the $RW_2$-nearest Gaussian, and the dashed curve represents the moment-matching Gaussian. Although the two Gaussian approximations are close, they do not coincide, and the difference becomes larger as $m$ increases.
• Figure 3: $RW_2$ energy landscape over Gaussian directions. The contour plot shows the value of $W_2^2\!\bigl(\mu,\mathcal{N}(0,\Sigma(\lambda,\theta))\bigr)$ as a function of the eigenvalue parameter $\lambda$ and the rotation angle $\theta$. The red dot denotes the moment-matching Gaussian, while the red star denotes the $RW_2$-nearest Gaussian. The latter achieves a strictly smaller $RW_2$ distance, demonstrating that the moment-matching Gaussian does not lie on the nearest Gaussian ray.
• Figure 4: Relative Wasserstein angles for two-dimensional Gaussian mixture distributions with multiple centers arranged on a regular grid. Each panel shows empirical samples from a Gaussian mixture with $r \times c$ components, together with the corresponding $RW_2$ angle between the empirical distribution and its covariance-matching Gaussian. As the number of mixture components increases, the Wasserstein angle increases, indicating growing deviation from Gaussianity due to multimodality.

Theorems & Definitions (16)

• Definition 2.1: $p$-norm Optimal Transport Problem villani2003topics
• Definition 2.2: Wasserstein Distance villani2009OT
• Definition 2.3: Relative Translation Invariant Wasserstein Distance
• Definition 3.1: Relative Wasserstein Angle
• Theorem 3.2
• Definition 3.3: Projection Distance
• Proposition 3.4: Closed-Form Projection Distance and $RW_2$ Angle in One Dimension
• Lemma 1.1: Uniqueness of the Optimal Transport Map
• proof
• Lemma 1.2: Form of Geodesics