Light Entropic Optimal Transport on Riemannian Manifolds

2026-10-02

AI Summary

This study addresses the high computational cost and underutilization of geometric structure in entropy-regularized optimal transport (OT) on Riemannian manifolds. To this end, we propose ManifoldLightOT, a method that constructs geometry-adapted Gibbs kernels with parameterized potential functions, enabling closed-form normalization and direct sampling without iterative optimization or dynamic simulation. The approach is compatible with manifold structures such as spheres and SO(3). Its core contribution lies in deeply integrating Riemannian geometry with kernel methods to provide an efficient closed-form solution for entropic OT on manifolds. Experiments on both synthetic and real-world datasets demonstrate that ManifoldLightOT significantly outperforms existing manifold OT baselines while effectively preserving the capability for direct sampling.

Abstract

Entropic Optimal Transport (EOT) has become a practical framework for learning stochastic couplings between complex distributions, with applications in generative modeling and domain adaptation. However, most EOT solvers are designed for Euclidean spaces, while manifold extensions remain limited and often rely on costly iterative methods, simulated dynamics, or generic neural models that do not fully exploit the underlying geometry. We introduce ManifoldLightOT, a light approach for learning kernel-induced EOT couplings directly on common manifolds. Using the kernel form of the EOT solution, we construct geometry-specific Gibbs kernels together with compatible potential parameterizations for spheres, tori, $\mathrm{SO}(3)$, and $\mathrm{SE}(3)$. These choices yield closed-form normalization and directly sampleable conditional distributions. Our formulation naturally extends to products of manifolds, making it applicable to more complex geometries. The parameters of the potentials are optimized directly from samples using Monte Carlo estimates of the learning objective. Through synthetic and real-world experiments, we show that ManifoldLightOT often outperforms existing manifold OT methods while retaining direct sampling.

Problem

• Entropic Optimal Transport
• Riemannian Manifolds
• Stochastic Couplings
• Manifold Geometry

Innovation

• Entropic Optimal Transport
• Riemannian Manifolds
• Gibbs Kernels
• ManifoldLightOT
• Direct Sampling