发表机构
KAIST（韩国科学技术院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
RW-Flow提出基于Wasserstein梯度流的一步生成框架，解决紧致流形上可辨识性问题，并在多个基准上优于现有一步方法。
AI 中文摘要
流形值数据及其所诱导的分布在许多领域普遍存在，范围从地震等地理空间事件的位置到编码三维结构信息的生物分子扭转角。尽管扩散和基于流的生成模型已成功扩展到紧致流形，但采样通常需要数十或数百次顺序网络评估。我们引入了RW-Flow，一个通过Wasserstein梯度流在紧致流形上学习一步生成模型的理论基础框架。主要挑战是可辨识性：将速度场驱动为零应确保模型分布与目标分布匹配。我们在紧致、连通的黎曼流形上建立了可辨识性的充分必要条件。我们特别证明了，对于对称的Lipschitz连续代价函数，由Sinkhorn散度诱导的速度场是可辨识的当且仅当相关的Gibbs核是非退化的。这一刻画为在紧致流形上设计可辨识代价函数提供了通用原则。它还揭示了平方测地距离，即平方欧氏距离在流形上的自然类比，并不总能保证可辨识性。在涉及地理空间事件、蛋白质侧链扭转角、RNA骨架扭转角以及离散为三角网格的一般流形的基准测试中，在公平比较条件下，RW-Flow在几乎所有设置中均优于现有的一步方法。
英文摘要
Manifold-valued data, and consequently the distributions they induce, are prevalent across many domains, ranging from the locations of geospatial events, such as earthquakes, to biomolecular torsion angles that encode information about three-dimensional structure. While diffusion and flow-based generative models have been successfully extended to compact manifolds, sampling typically requires tens or hundreds of sequential network evaluations. We introduce RW-Flow, a theoretically grounded framework for learning one-step generative models on compact manifolds via Wasserstein gradient flows. The main challenge is identifiability: driving the velocity field to zero should guarantee that the model distribution matches the target distribution. We establish a necessary and sufficient condition for identifiability on compact, connected Riemannian manifolds. We specifically show that, for a symmetric, Lipschitz-continuous cost function, the velocity field induced by the Sinkhorn divergence is identifiable if and only if the associated Gibbs kernel is nondegenerate. This characterization provides a general principle for designing identifiable costs on compact manifolds. It also reveals that the squared geodesic distance, the natural manifold analogue of the squared Euclidean distance, does not always guarantee identifiability. Across benchmarks involving geospatial events, protein side chain torsion angles, RNA backbone torsion angles, and general manifolds discretized as triangular meshes, RW-Flow outperforms existing one-step methods in nearly all settings under fair comparison conditions.
Comments27 pages
