# Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2606.07926 · HTML: https://mlsanity.com/paper/2606.07926/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 23 references that are in the map.
- First posted on arXiv 6 Jun 2026.

## Builds on (most cited first, 5 of 5)

- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [MOABB: Trustworthy algorithm benchmarking for BCIs](https://mlsanity.com/paper/1805.06427/) (2018)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Linearized optimal transport on manifolds](https://mlsanity.com/paper/2303.13901/) (2023)

## Built on by (most cited first, 1 of 1)

- [When Riemann flows with Wasserstein: Generative Modeling of Probability Distributions on Manifolds](https://mlsanity.com/paper/2609.25659/) (2026)

## Abstract (arXiv)

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 Fr'echet 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 Fr'echet 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 Fr'echet 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.