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, are inherently unable to scale to high dimensions, or use approximations for limiting quantities that result in biased training objectives. Riemannian Flow Matching bypasses these limitations and offers several advantages over previous approaches: it is simulation-free on simple geometries, does not require divergence computation, and computes its target vector field in closed-form. The key ingredient behind RFM is the construction of a relatively simple premetric for defining target vector fields, which encompasses the existing Euclidean case. To extend to general geometries, we rely on the use of spectral decompositions to efficiently compute premetrics on the fly. Our method achieves state-of-the-art performance on real-world non-Euclidean datasets, and we demonstrate tractable training on general geometries, including triangular meshes with highly non-trivial curvature and boundaries.

翻译：我们提出了黎曼流匹配（RFM），这是一个简单而强大的框架，用于在流形上训练连续归一化流。现有的流形上生成建模方法要么需要昂贵的模拟，本质上无法扩展到高维，要么使用导致训练目标有偏的近似极限量。黎曼流匹配绕过了这些限制，并相较于先前方法具有多项优势：在简单几何上无需模拟，不需要计算散度，且能以闭式形式计算目标向量场。RFM的关键在于构建一个相对简单的预度量来定义目标向量场，这涵盖了已有的欧几里得情形。为了扩展到一般几何，我们利用谱分解来高效地实时计算预度量。我们的方法在真实世界的非欧几里得数据集上达到了最先进的性能，并展示了在一般几何（包括具有高度非平凡曲率和边界的三角网格）上的可处理训练。