Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds
Researchers have developed a mathematical framework for converting probabilistic optimal transport couplings into deterministic maps on Riemannian manifolds, where curvature complicates standard Euclidean methods. The work introduces an intrinsic projection based on conditional Fréchet means and a tangential log-exp projection, along with a new 'conditional-variance Monge defect' measure. The framework addresses a gap in machine learning pipelines that require deterministic maps when working with non-Euclidean data such as EEG covariance matrices.
A preprint posted to arXiv presents a theoretical and computational framework for barycentric projections of optimal transport couplings on Riemannian manifolds. In Euclidean space, barycentric projection converts a probabilistic coupling into a deterministic map via conditional expectations, but curvature and cut loci on manifolds make this operation substantially more complex. The authors introduce an intrinsic projection that maps each source point to the conditional Fréchet mean of its destination distribution, proving it is the optimal deterministic representative under squared geodesic loss. A complementary tangential log-exp projection is also developed and shown to be exact in Euclidean settings, compatible with Brenier-McCann maps in the Monge case, and interpretable as a single Riemannian gradient step. The minimum value of the intrinsic objective yields an integrated conditional Fréchet variance, which the authors term the conditional-variance Monge defect, vanishing precisely when the coupling is map-induced. Experiments on spherical data, synthetic symmetric positive definite matrices, and real EEG covariance data validate the proposed division of roles between the two projections. The work was submitted on June 6, 2026, and has not yet undergone formal peer review.
What's missing
As a preprint, this work has not yet been peer-reviewed. The abstract does not discuss runtime complexity or convergence guarantees for the discrete algorithms. The generality of the framework to manifolds with non-unique geodesics or highly irregular cut loci is not addressed in the abstract.
What different sources said
- arXiv cs.LGCenter
Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds
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