发表机构
Applied AI Institute（应用人工智能研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出ManifoldLightOT，一种在常见黎曼流形上利用核形式构建熵最优传输耦合的轻量方法，通过几何特定吉布斯核实现闭式归一化与直接采样，并在实验中优于现有流形OT方法。
AI 中文摘要
熵最优传输（EOT）已成为学习复杂分布之间随机耦合的实用框架，在生成建模和域适应中具有应用。然而，大多数EOT求解器是为欧几里得空间设计的，而流形扩展仍然有限，且通常依赖于昂贵的迭代方法、模拟动力学或不能充分利用底层几何的通用神经模型。我们引入了ManifoldLightOT，一种直接在常见流形上学习核诱导EOT耦合的轻量方法。利用EOT解的核形式，我们为球面、环面、SO(3)和SE(3)构建了特定几何的吉布斯核以及兼容的势参数化。这些选择产生了闭式归一化和可直接采样的条件分布。我们的公式自然扩展到流形的乘积，使其适用于更复杂的几何。势的参数通过使用学习目标的蒙特卡洛估计直接从样本中优化。通过合成和真实世界实验，我们表明ManifoldLightOT在保留直接采样的同时，通常优于现有的流形OT方法。
英文摘要
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.
