Lee Cheuk-Kit, Florentin Coeurdoux, Yuyuan Chen, Sophia Tang, Peter Potaptchik, Yilun Du, Michael Samuel Albergo, Eric Vanden-Eijnden
原文摘要
We propose an instantiation of flow matching that relies on a time-independent velocity field---an autonomous flow---to exactly map between two distributions when the target is supported on a sufficiently lower-dimensional data manifold. We also show that the associated one-step generative map is constant in the direction of the autonomous velocity field and therefore satisfies a simple conservation equation, which motivates learning the map directly from samples. These autonomous flows and maps give a dynamical meaning to the flux constraint of Beckmann's transportation problem, and we show that, under suitable conditions, the construction extends beyond flow matching to other probability fluxes satisfying the same constraint. The resulting framework includes, for instance, a closed-form Coulomb transport closely related to Poisson Flow and a flow-matching-consistent version of Equilibrium Matching. We illustrate these distinctions and demonstrate the effectiveness of the autonomous flow and the one-step map on ImageNet $256\times256$.
Zhengkai Pan, Peter Potaptchik, Wenxi Yao, Michael S. Albergo, Jakiw Pidstrigach
原文摘要
Recent one-step generative models accelerate sampling by learning deterministic flow maps of the underlying dynamics. These methods rely on learning from ordinary differential equations, leaving open how to define an exact distillation procedure for stochastic dynamics. We introduce the Itô map, an any-step stochastic flow map that takes an intermediate state and Brownian path and predicts future states in a single pass. The Itô map formulation yields novel estimators for inference-time control by providing cheap, differentiable access to posterior samples. Empirically, Itô maps produce diverse, conditionally valid endpoint samples from fixed intermediate states and support strong steering performance on synthetic and image-generation benchmarks. These results establish any-step SDE integration as a useful primitive for posterior sampling and stochastic control.
Tyler Farghly, Peter Potaptchik, Samuel Howard, George Deligiannidis, Jakiw Pidstrigach
原文摘要
Diffusion models have achieved state-of-the-art performance, demonstrating remarkable generalisation capabilities across diverse domains. However, the mechanisms underpinning these strong capabilities remain only partially understood. A leading conjecture, based on the manifold hypothesis, attributes this success to their ability to adapt to low-dimensional geometric structure within the data. This work provides evidence for this conjecture, focusing on how such phenomena could result from the formulation of the learning problem through score matching. We inspect the role of implicit regularisation by investigating the effect of smoothing minimisers of the empirical score matching objective. Our theoretical and empirical results confirm that smoothing the score function -- or equivalently, smoothing in the log-density domain -- produces smoothing tangential to the data manifold. In addition, we show that the manifold along which the diffusion model generalises can be controlled by choosing an appropriate smoothing.