Exploiting Low Frequencies as a Prior for Optimal Transport Flows

Optimal transport (OT) couplings between prior and data straighten flow matching trajectories and enable fast, even single-step, generation. But computing OT is cubic in the number of samples and existing methods solve the problem approximately at the cost of persistent bias or significant overhead. Instead of treating the prior distribution (typically, a Gaussian) as a given, we treat it as a design choice. The question now is what prior distribution simplifies OT coupling? There might be many different priors that achieve this, leaving us free to choose one that is also tractable to sample. We identify low-frequency projection of natural images as such a choice, and show that the identity coupling between data and its low-frequency representation is OT-optimal for nearly all sampled pairs. The prior is structured enough to be sampled by a lightweight model at inference, and the remaining flow-matching task reduces to synthesizing high-frequency detail. The criterion admits more than the deterministic projection. Interpolating the prior with Gaussian noise improves generation quality and the identity coupling remains optimal. The approach requires no modifications to the flow model itself, and integrates naturally with latent-space models, classifier-free guidance, and one-step generation frameworks. Our method reduces trajectory curvature by more than $2\times$ relative to the strongest baseline on each benchmark.