A Unified Framework for Data-Free One-Step Sampling via Wasserstein Gradient Flows

arXiv:2605.17808v1 Announce Type: new Abstract: We develop a unified theoretical framework for data-free one-step sampling from unnormalized target distributions based on Wasserstein gradient flows. For a broad class of standard f-divergence objectives, we show that the induced velocity field admits the universal form mathbf{V}(x)=w(r(x)),eta(x), where eta(x)=nabla log (p(x)/q(x)) is shared across objectives and w is determined solely by the choice of divergence. This decomposition shows that standard f-divergence drifts share the same asymptotic target distribution p and differ primarily in how they redistribute transient repair effort across under-covered regions. To formalize this distinction, we derive a one-step regional-response theory for a soft under-coverage functional and obtain a compression--elasticity identity that links divergence choice to the geometry of mass transport into under-covered regions. We further extend the framework beyond the f-divergence family to the Log-Variance (LV) divergence, analyze how the reference distribution alters the resulting drift structure, and motivate a practical LV-inspired surrogate for data-free training. Based on this theory, we instantiate the framework with a KDE-based implementation and describe a complementary normalizing-flow route, enabling one-step inference after training. Experiments on multimodal Gaussian-mixture benchmarks are consistent with the theoretical predictions and demonstrate effective one-step sampling on these targets.

Source: arXiv cs.LG | 2026-05-19