Learning Manifold Data with Flow Matching

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. These bounds meet near-minimax rates for flow-matching transformers of any order. Our results show how flow-matching transformers escape the curse of dimensionality by utilizing intrinsic data structure.