Learning Manifold Data with Flow Matching

Sophia Pi, Mingcheng Lu, Jerry Yao-Chieh Hu, Maojiang Su, Weimin Wu, Han Liu

Keywords: flow matching

Abstract: We study flow-matching transformers when data lie on a low-dimensional manifold. Our key insight is a flow decomposition that splits motion along the manifold from motion off the manifold. The scheme works for first- and higher-order flow matching and ties model complexity to the intrinsic manifold dimension. Building on these, we establish tighter sample-complexity bounds for velocity approximation, velocity estimation, and distribution estimation. Our results show how flow-matching transformers escape the curse of dimensionality by utilizing intrinsic data structure.

Submission Number: 97