Learning Manifold Data with Flow Matching

We study statistical rates of flow-matching transformers when data lie on a low-dimensional manifold. Our key insight is a velocity decomposition that splits motion along the manifold from motion off the manifold. The scheme works for firstand higher-order flow matching and ties 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 data structure.