On Closed-Form Couplings

Tobias Höppe, Stefan Bauer, qiang liu, Andrea Dittadi, Kirill Neklyudov

Keywords: Diffusion, Flow Matching, Rectified Flow, (semi-discrete) couplings

TL;DR: We study closed-form probability-flow dynamics as an analytic teacher for rectified flows, and show why they fail to provide a reliable training signal.

Abstract: Few-step generative modelling is an open challenge for flow models. Rectified flows tackle it by distilling a pre-trained “teacher” into a few-step “student”, using strong noise–data couplings supplied by the teacher. For a finite dataset and a Gaussian probability path, the probability-flow vector field induced by the empirical distribution is available in closed form, which would allow us to skip training a teacher model. Surprisingly, these couplings turn out to be poor teachers and significantly reduce the performance of the student. We analyse this phenomenon empirically and theoretically, arguing that it stems from intrinsic ambiguity in the induced couplings caused by the strong sensitivity of terminal states to small initialisation perturbations. Under symmetry assumptions, we further prove that the closed-form probability-flow vector field preserves dataset symmetries and induces invariant Voronoi partitions.

Submission Number: 47