An Error Analysis of Flow Matching for Deep Generative Modeling

Continuous Normalizing Flows (CNFs) have proven to be a highly efficient technique for generative modeling of complex data since the introduction of Flow Matching (FM). The core of FM is to learn the constructed velocity fields of CNFs through deep least squares regression. Despite its empirical effectiveness, theoretical investigations of FM remain limited. In this paper, we present the first end-to-end error analysis of CNFs built upon FM. Our analysis shows that for general target distributions with bounded support, the generated distribution of FM is guaranteed to converge to the target distribution in the sense of the Wasserstein-2 distance. Furthermore, the convergence rate is significantly improved under an additional mild Lipschitz condition of the target score function.

While Flow Matching has shown great results in practice, scientists didn't have a strong theoretical understanding of why it works so well or guarantees about how close the generated data can get to the real data. This research paper provides the first complete theoretical analysis of how errors can build up when using Flow Matching, from the initial training data all the way to the final generated data.The researchers proved that for most types of complex data (as long as the data doesn't spread out infinitely, which they call "bounded support"), the data generated by a model trained with Flow Matching is guaranteed to get closer and closer to the real data. We use a mathematical measuring called the "Wasserstein-2 distance" to show this convergence.In essence, this paper mathematically confirms that Flow Matching is a solid technique for teaching computers to generate complex, realistic data, and it even explains how certain properties of the data can make the learning process more efficient.