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. Our results show how flow-matching transformers escape the curse of dimensionality by utilizing intrinsic data structure.

Modern generative AI systems are capable of creating images, videos, and other complex data, even though such data are often described in millions of parameters. In practice, however, the meaningful variation in such data is often much simpler than it appears. This paper focus on a training method called flow matching, which learns how to gradually transform simple random noise into realistic data. Our main finding is that, when the data actually vary along only a small number of meaningful directions, the learned transformation can be separated into two parts: one that moves data within this simpler structure, and one that pulls points back toward it. This separation lets us prove that flow-matching models with transformer networks can learn using resources that depend mainly on the true underlying complexity of the data, rather than the much larger number of raw input dimensions.