We propose Riemannian Flow Matching (RFM), a simple yet powerful framework for training continuous normalizing flows on manifolds. Existing methods for generative modeling on manifolds either require expensive simulation, inherently cannot scale to high dimensions, or use approximations to limiting quantities that result in biased objectives. Riemannian Flow Matching bypasses these inconveniences and exhibits multiple benefits over prior approaches: It is completely simulation-free on simple geometries, it does not require divergence computation, and its target vector field is computed in closed form even on general geometries. The key ingredient behind RFM is the construction of a simple kernel function for defining per-sample vector fields, which subsumes existing Euclidean cases. Extending to general geometries, we rely on the use of spectral decompositions to efficiently compute kernel functions. Our method achieves state-of-the-art performance on real-world non-Euclidean datasets, and we showcase, for the first time, tractable training on general geometries, including on triangular meshes and maze-like manifolds with boundaries.

翻译：我们提出黎曼流匹配（Riemannian Flow Matching, RFM），这是一个在流形上训练连续归一化流的简洁而强大的框架。现有基于流形的生成建模方法或需昂贵模拟计算，或本质上难以扩展至高维空间，或采用有偏目标量的近似处理。黎曼流匹配规避了上述缺陷，并展现出相较于先前方法的多个优势：在简单几何结构上完全无需模拟计算，无需计算散度，即使在一般几何结构上也能以闭式解形式获得目标向量场。RFM的核心在于构建简单核函数以定义逐样本向量场，该函数统一涵盖了现有欧几里得情形。为扩展至一般几何结构，我们借助谱分解方法实现核函数的高效计算。该方法在真实非欧几里得数据集上取得了最先进性能，并首次展示了在一般几何结构（包括三角网格和带边界迷宫状流形）上的可训练性。