Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds
Researchers developed a mathematical framework for converting probabilistic optimal transport couplings into deterministic maps on curved spaces (Riemannian manifolds), extending techniques previously available only in flat Euclidean space. The work introduces two projection methods—an intrinsic projection based on conditional Fréchet means and a tangential log-exp projection—with theoretical guarantees and practical applications. This advancement is important because many machine learning pipelines require deterministic mappings, and the framework enables optimal transport methods to work on non-Euclidean geometries common in real-world data.
The paper addresses a fundamental challenge in optimal transport theory: while optimal transport couplings are naturally probabilistic objects, many machine learning applications require deterministic maps. In Euclidean space, barycentric projection solves this by taking conditional expectations, but this approach breaks down on curved Riemannian manifolds due to geometric complications from curvature and cut loci. The authors develop a complete framework with two complementary approaches: an intrinsic projection that maps each source point to the conditional Fréchet mean of its destination distribution (proven optimal under squared geodesic loss), and a tangential log-exp projection that serves as a local approximation. The intrinsic projection's optimality is characterized by an integrated conditional Fréchet variance—a new conditional-variance Monge defect that equals zero precisely for map-induced couplings. The authors validate their theory with experiments on spherical data, synthetic symmetric positive definite matrices, and real EEG covariance data, demonstrating practical utility across different geometric settings.
What's missing
The paper does not discuss computational complexity or scalability of the proposed algorithms for high-dimensional manifolds, nor does it compare runtime performance against alternative approaches for converting couplings to maps.
What different sources said
- arXiv cs.LGCenter
Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds
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