Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds

2026-06-05

AI Summary

This work addresses the challenge of efficiently converting probabilistic optimal transport couplings into deterministic maps on Riemannian manifolds to better suit learning tasks. To this end, the authors propose an intrinsic barycentric projection framework that defines an optimal deterministic representative via conditional Fréchet means under a geodesic squared loss, and introduces the Monge defect to quantify its deviation from the true Brenier–McCann map. They further develop a compatible log–exp local projection method in the tangent space that aligns with this deterministic mapping. Leveraging tools from Fréchet means, geodesic distances, and Riemannian gradient optimization, theoretical analysis and experiments on spherical data, SPD matrices, and EEG covariance manifolds demonstrate the variational optimality of the intrinsic projection and the local displacement efficacy of the tangent-space projection.

Abstract

Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps. In Euclidean space, barycentric projection converts a coupling into a map by taking conditional expectations, but on a Riemannian manifold curvature and cut loci make this operation nontrivial. We develop a framework for barycentric projections of transport couplings on Riemannian manifolds. The intrinsic projection maps each source point to the conditional Fréchet mean of its destination law and is shown to be the best deterministic representative under squared geodesic loss. The corresponding minimum value is an integrated conditional Fréchet variance, which vanishes exactly for map-induced couplings and therefore defines a conditional-variance Monge defect. We also study a tangential log-exp projection, prove its Euclidean exactness, its compatibility with Brenier-McCann maps in the Monge case, and its interpretation as the first unit Riemannian gradient update for the intrinsic objective. For discrete couplings, both constructions decompose row-wise into weighted Fréchet mean and log-exp problems. Experiments on spherical data, synthetic SPD data, and real EEG covariance matrices support the proposed division of roles: the intrinsic projection is the variational representative, while the tangential projection is a useful local displacement surrogate.

Problem

• optimal transport
• barycentric projection
• Riemannian manifolds
• deterministic maps
• Fréchet mean

Innovation

• barycentric projection
• Riemannian manifold
• optimal transport
• Fréchet mean
• Monge defect