Concentrating Velocities and Accelerations in Hierarchical Flow Matching

Hierarchical rectified flow models the multimodal velocity distribution of a flow model by learning an acceleration field, replacing a single ODE with a nested pair. Nothing in this construction forces the distribution modeled at one level to be simpler than the one above. Specifically, at $t=0$, when source and target samples are paired independently, the velocity distribution there is exactly a shifted copy of the data distribution. We address this by applying mini-batch optimal transport couplings within each level, giving two methods at different operating points. Coupling data pairs concentrates the velocity distribution and straightens the outer trajectories. Coupling velocity pairs concentrates the acceleration distribution and straightens the inner trajectories. Neither alone suffices, because the sampling cost is the product of the steps taken at each level: data coupling alone leaves the inner solve expensive, while velocity coupling alone leaves the outer trajectories curved. Applying both makes every level cheap at once, which is what enables generation with small number of neural function evaluations. We prove that data and velocity couplings hierarchically simplify the velocity and acceleration distributions. We also demonstrate that the couplings improve generation quality and efficiency on synthetic data, MNIST, CIFAR-10, CelebA-HQ 256, and ImageNet 256.