Title: Designing Optimal Transport Flow Matching

Abstract

Flow matching algorithms are designed to move samples from a straightforward prior distribution toward a complex target data distribution. When the pairing between prior and data points is governed by optimal transport (OT), the resulting trajectories become straight and non-intersecting. This geometric property allows for accelerated generation, potentially even in a single step. Nevertheless, calculating the OT coupling in high-dimensional spaces remains computationally prohibitive. Current approaches attempt to approximate this coupling, but they often introduce lasting bias or incur substantial computational costs.

Instead of attempting to solve for the OT coupling directly, this work proposes a fundamental reformulation of the problem. By shifting the perspective to treat the prior as a configurable design element rather than a static input, the uniqueness of the OT coupling between the prior and the data is removed. This flexibility allows us to select a prior that possesses an OT-optimal identity coupling with the data while remaining easy to sample. We demonstrate that the low-frequency projection of natural images serves as an ideal candidate for this purpose.

The identity coupling between the original data and its low-frequency approximation is shown to be empirically OT-optimal. This specific prior is sufficiently structured to be sampled efficiently by a lightweight model during inference. Consequently, the remaining flow-matching challenge is simplified to the synthesis of high-frequency details. Furthermore, blending the prior with Gaussian noise enhances generation quality without compromising the OT coupling structure. This methodology requires no changes to the underlying flow model architecture and fits seamlessly into latent-space models, classifier-free guidance mechanisms, and one-step generation frameworks. Evaluation across various benchmarks indicates that our approach decreases trajectory curvature by a factor of more than two compared to standard flow matching techniques, resulting in superior generation quality, particularly in few-step scenarios.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC