PathSpeed: Accelerating Structural Recovery in Flow Matching

Learning a generative distribution requires recovering its modes, connected components, and loops. We study how training-time sampling can accelerate this recovery in flow matching. PathSpeed measures the motion of path marginals before velocity training and uses it to construct a fixed training-time distribution. We estimate squared path speed from adjacent marginal samples, then smooth and standardize the estimates. We show that this finite-difference measurement splits exactly into changes in projected means and scales and changes in standardized shape. In the continuous limit, squared path speed measures the energy of the conditional-average motion that transports the projected marginals. Integrated speed bounds changes in persistence diagrams at a fixed observation scale. On controlled distributions, PathSpeed reaches the final structural error of uniform time sampling (Uniform) with 33–49% fewer updates and achieves lower final errors. Ablations show that shape information and the placement of sampling probabilities along time both contribute. On CIFAR-10 and class-conditional latent ImageNet-256, PathSpeed obtains the lowest mean FID among the compared schedules and reaches Uniform's final FID with 42% and 52% fewer updates, respectively. The cached distribution adds no velocity-network forward or backward pass during training.