Title: OAT-FM: Optimal Acceleration Transport for Improved Flow Matching
Primary area: generative models
Keywords: Flow Matching, Optimal Transport

Abstract: As a powerful technique in generative modeling, Flow Matching (FM) aims to learn velocity fields from noise to data, which is often explained and implemented as solving Optimal Transport (OT) problems. In this study, we bridge FM and the recent theory of Optimal Acceleration Transport (OAT), developing an improved FM method called OAT-FM and exploring its benefits in both theory and practice. 
In particular, we demonstrate that the straightening objective hidden in existing OT-based FM methods is mathematically equivalent to minimizing the physical action associated with acceleration defined by OAT.  Accordingly, instead of enforcing constant velocity, OAT-FM optimizes the acceleration transport in the product space of sample and velocity, whose objective corresponds to a necessary and sufficient condition of flow straightness. An efficient algorithm is designed to achieve OAT-FM with low complexity. OAT-FM motivates a new two-phase FM paradigm: Given a generative model trained by an arbitrary FM method, whose velocity information has been relatively reliable, we can fine-tune and improve it via OAT-FM. This paradigm eliminates the risk of data distribution drift and the need to generate a large number of noise data pairs, which consistently improves model performance in various generative tasks.

TL;DR: We develop a new flow matching method based on the theory of optimal acceleration transport, leading to a new two-phase FM paradigm with theoretical guarantees.