Procrustes Wasserstein Metric: A Modified Benamou-Brenier Approach with Applications to Latent Gaussian Distributions
Kevine Meugang Toukam
TL;DR
The paper introduces the Procrustes Wasserstein distance, a modified Benamou-Brenier framework that enforces global isometry invariance by optimizing over orthogonal transformations, reducing the problem to orthogonal equivariance and centering. It provides a static formulation and a complete Gaussian case: the Procrustes-Wasserstein distance between Gaussians equals the squared Euclidean distance between the ordered square roots of their covariance eigenvalues, with the optimal rotation given by a simple eigenbasis alignment. A quotient-space perspective is developed, showing completeness and geodesic structure in the Procrustes setting, and the Gaussian case yields an explicit, computable metric. The framework is then applied to recover latent Gaussian distributions from observed, orthogonally transformed data, deriving asymptotically unbiased estimators for the latent covariance up to an orthogonal transformation, with the Fréchet mean under PW metric providing the estimator. Altogether, the work connects optimal transport invariance principles with practical latent-distribution recovery under unknown orthogonal nuisances.
Abstract
We introduce a modified Benamou-Brenier type approach leading to a Wasserstein type distance that allows global invariance, specifically, isometries, and we show that the problem can be summarized to orthogonal transformations. This distance is defined by penalizing the action with a costless movement of the particle that does not change the direction and speed of its trajectory. We show that for Gaussian distribution resume to measuring the Euclidean distance between their ordered vector of eigenvalues and we show a direct application in recovering Latent Gaussian distributions.