Riemannian Neural Optimal Transport

2026-02-03

AI Summary

Existing neural optimal transport methods are largely confined to Euclidean spaces and rely on discretization, suffering from the curse of dimensionality when extended to high-dimensional Riemannian manifolds. This work proposes the first continuous neural parameterization framework that directly constructs optimal transport maps on Riemannian manifolds, inherently respecting their geometric structure and circumventing discretization. By employing a geometry-aware neural network architecture, the method enables continuous modeling of Riemannian optimal transport maps with approximation complexity growing sub-exponentially in dimension, thereby significantly mitigating the curse of dimensionality. Experiments demonstrate that the approach consistently outperforms existing discretization-based baselines on both synthetic and real-world datasets, achieving substantial improvements in scalability and performance.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Computational optimal transport (OT) offers a principled framework for generative modeling. Neural OT methods, which use neural networks to learn an OT map (or potential) from data in an amortized way, can be evaluated out of sample after training, but existing approaches are tailored to Euclidean geometry. Extending neural OT to high-dimensional Riemannian manifolds remains an open challenge. In this paper, we prove that any method for OT on manifolds that produces discrete approximations of transport maps necessarily suffers from the curse of dimensionality: achieving a fixed accuracy requires a number of parameters that grows exponentially with the manifold dimension. Motivated by this limitation, we introduce Riemannian Neural OT (RNOT) maps, which are continuous neural-network parameterizations of OT maps on manifolds that avoid discretization and incorporate geometric structure by construction. Under mild regularity assumptions, we prove that RNOT maps approximate Riemannian OT maps with sub-exponential complexity in the dimension. Experiments on synthetic and real datasets demonstrate improved scalability and competitive performance relative to discretization-based baselines.

Problem

• Riemannian manifolds
• optimal transport
• curse of dimensionality
• neural networks
• generative modeling

Innovation

• Riemannian Neural Optimal Transport
• curse of dimensionality
• continuous transport maps
• geometric deep learning
• optimal transport on manifolds