# Entropic Riemannian Neural Optimal Transport

arXiv: https://arxiv.org/abs/2605.04255 · HTML: https://mlsanity.com/paper/2605.04255/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 28 references that are in the map.
- First posted on arXiv 5 May 2026.

## Builds on (most cited first, 10 of 10)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Non-Euclidean Universal Approximation](https://mlsanity.com/paper/2006.02341/) (2020)
- [Semi-dual Regularized Optimal Transport](https://mlsanity.com/paper/1811.05527/) (2018)
- [Implicit Riemannian Concave Potential Maps](https://mlsanity.com/paper/2110.01288/) (2021)
- [Riemannian Neural Optimal Transport](https://mlsanity.com/paper/2602.03566/) (2026)

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

- [Light Entropic Optimal Transport on Riemannian Manifolds](https://mlsanity.com/paper/2610.03085/) (2026)

## Abstract (arXiv)

Many machine learning problems involve data supported on curved spaces such as spheres, rotation groups, hyperbolic spaces, and general Riemannian manifolds, where Euclidean geometry can distort distances, averages, and the resulting optimal transport (OT) problem. Existing manifold OT methods have pursued amortized out-of-sample maps, while entropic regularization has made discrete OT more scalable, but these advantages have remained largely disjoint. We propose Entropic Riemannian Neural Optimal Transport (Entropic RNOT), a unified framework that combines intrinsic entropic OT with amortized out-of-sample evaluation on Riemannian manifolds. Our method learns a single target-side Schr"odinger potential through a neural pullback parameterization, recovers the induced Gibbs coupling, and uses the resulting conditional laws to construct intrinsic transport surrogates. These include barycentric projections on Cartan-Hadamard manifolds and heat-smoothed conditional surrogates on stochastically complete manifolds, the latter turning possibly atomic target laws into absolutely continuous ones. For fixed regularization varepsilon>0, we prove that the proposed hypothesis class recovers the entropic optimal coupling in strong probabilistic metrics. As consequences, barycentric surrogates converge in L2, while heat-smoothed surrogates are stable at fixed heat time and asymptotically unbiased as the heat time vanishes. The guarantees hold for compactly supported data on possibly noncompact manifolds. Empirically, our method matches or improves over Euclidean, tangent-space, and log-Euclidean baselines on benchmarks over mathbbS2, SO(3), SPD(3), SE(3), and mathbbH2, scales favorably relative to discrete manifold Sinkhorn, and in a protein-ligand docking application, refines poses on SE(3) without retraining or per-instance optimization.