提出连续神经最优传输方法，解决高维流形上生成建模的维度灾难问题。
Riemannian Neural Optimal Transport
- 用连续神经网络参数化流形上的传输映射，避免离散化
- 理论证明复杂度为亚指数级，突破维度诅咒
- 在合成与真实数据上表现优于传统离散方法
计算最优传输（OT）为生成建模提供了严谨框架。神经OT方法利用神经网络以摊销方式从数据中学习OT映射（或势函数），训练后可进行样本外评估，但现有方法仅适用于欧氏几何。将神经OT推广至高维黎曼流形仍是开放挑战。本文证明：任何在流形上生成传输映射离散近似的OT方法必然面临维度诅咒——达到固定精度所需的参数量随流形维度呈指数增长。针对此限制，我们提出黎曼神经最优传输（RNOT）映射，其为流形上连续的神经网络参数化，避免离散化并天然融合几何结构。在弱正则性假设下，我们证明RNOT映射以亚指数复杂度逼近黎曼OT映射。在合成与真实数据集上的实验表明，该方法具有更好的可扩展性和与基于离散化的基线相当的性能。
原文摘要 · Abstract (English)
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.
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