Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds

2026-06-06 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

A new framework addresses the challenge of converting probabilistic optimal transport couplings into deterministic maps on Riemannian manifolds, where curvature and cut loci complicate standard Euclidean barycentric projections. This framework introduces an intrinsic projection, which maps each source point to the conditional Fréchet mean of its destination law, serving as the optimal deterministic representative under squared geodesic loss. The minimum value of this projection defines a conditional-variance Monge defect, vanishing for map-induced couplings. Additionally, a tangential log-exp projection is developed, demonstrating Euclidean exactness and compatibility with Brenier-McCann maps. For discrete couplings, both methods decompose into weighted Fréchet mean and log-exp problems. Experiments on spherical data, synthetic SPD data, and real EEG covariance matrices validate the intrinsic projection as a variational representative and the tangential projection as a local displacement surrogate.

Key takeaway

For research scientists developing machine learning models on non-Euclidean data, you should consider integrating barycentric projections to convert probabilistic optimal transport couplings into deterministic maps. This framework provides both a variational intrinsic projection for optimal representation and a tangential log-exp projection for local displacement, offering robust tools for handling complex data structures like spherical or SPD matrices, thereby improving model interpretability and performance in Riemannian manifold settings.

Key insights

A framework enables deterministic map extraction from optimal transport couplings on Riemannian manifolds using intrinsic and tangential projections.

Principles

• Intrinsic projection is the best deterministic representative under squared geodesic loss.
• Conditional Fréchet variance defines a Monge defect for map-induced couplings.
• Tangential log-exp projection offers a useful local displacement surrogate.

Method

The framework involves computing conditional Fréchet means for intrinsic projection and utilizing log-exp operations for tangential projection, decomposing row-wise for discrete couplings.

In practice

• Apply intrinsic projection for variational representation in learning pipelines.
• Use tangential log-exp projection as a local displacement surrogate.
• Decompose discrete couplings into weighted Fréchet mean problems.

Topics

• Optimal Transport
• Riemannian Manifolds
• Barycentric Projections
• Fréchet Mean
• Geodesic Loss
• Machine Learning

Best for: AI Scientist, Research Scientist

Related on AIssential

• Static and Dynamic Approaches to Computing Barycenters of Probability Measures on Graphs — A dynamic optimal transport framework enables robust barycenter computation and analysis for probability measures on graphs.
• Geodesic Flow Matching for Denoising High-Dimensional Structured Representations — Geodesic Flow Matching effectively denoises Spatial Semantic Pointers by respecting their toroidal manifold geometry, preventing structural degradation.
• Geometric structures and deviations on James' symmetric positive-definite matrix bicone domain — New Finsler and dual information-geometric structures on SPD bicone domain offer alternative dissimilarity measures.
• On the Regularity and Generalization of One-Step Wasserstein-guided Generative Models for PDE-Induced Measures — The optimal transport map for PDE-induced measures is Hölder continuous, justifying one-step generative models.
• Horospherical Depth and Busemann Median on Hadamard Manifolds — Horospherical depth offers an intrinsic, robust statistical depth for Hadamard manifolds, defining a unique Busemann median.
• Are Common Substructures Transferable? Riemannian Graph Foundation Model with Neural Vector Bundles — Transferable graph substructures are linked to intrinsic geometry, addressed by GAUGE's Riemannian approach.

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.