Wasserstein Mahalanobis Distances for Recovering Latent Geometry

2026-08-06

AI Summary

This work addresses the recovery of latent geometric structure from nonlinear observations in the space of probability measures. It introduces, for the first time, the Wasserstein Mahalanobis distance—a geometry-aware metric between distributions constructed by combining optimal transport displacement fields with covariance operators on tangent spaces. This approach establishes a covariance-adaptive geometric framework for distribution-valued data, which exactly recovers the Mahalanobis distance of latent variables under affine transformations and achieves controllable error under general smooth transformations. Both theoretical analysis and numerical experiments demonstrate that the proposed distance effectively reconstructs the underlying latent geometry in the context of nonlinear independent component analysis.

Abstract

The Mahalanobis distance is a fundamental covariance-adapted metric for multivariate data and plays a central role in recovering latent geometry from nonlinear observations. We extend this principle from vector-valued data to probability measures by introducing a Wasserstein Mahalanobis distance. Our construction replaces Euclidean displacement vectors with optimal transport displacement fields and local covariance matrices with covariance operators defined on Wasserstein tangent spaces. We show that this construction inherits the geometry-recovery property underlying nonlinear independent component analysis. In particular, for Gaussian measures with common covariance transformed by a smooth nonlinear pushforward, the proposed Wasserstein Mahalanobis distance approximates the classical Mahalanobis distance between the transformed latent means. The correspondence is exact for affine transformations and holds up to controlled higher-order error terms for general smooth transformations. These results establish a distribution-valued analog of classical Mahalanobis geometry and provide theoretical support for covariance-adapted learning directly in Wasserstein space. Numerical experiments confirm the theoretical predictions and demonstrate accurate recovery of latent geometric structure.

Problem

• Wasserstein distance
• Mahalanobis distance
• latent geometry
• nonlinear observations
• probability measures

Innovation

• Wasserstein Mahalanobis distance
• optimal transport
• latent geometry recovery
• covariance operators
• nonlinear independent component analysis