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

2026-05-17

AI Summary

This work addresses the problem of one-step sampling from an unnormalized target distribution in the absence of data. Building upon Wasserstein gradient flows, the authors propose a unified framework that decomposes the velocity field into a general structure encompassing various f-divergences as well as the Log-Variance divergence. Theoretically, they demonstrate that drift fields derived from different divergences share the same asymptotic target distribution. By introducing a regional response theory and a compression–elasticity identity, they incorporate the Log-Variance divergence into this framework for the first time and formulate a practical surrogate objective. Empirical evaluation on multimodal Gaussian mixture benchmarks confirms the method’s effectiveness, achieving efficient and accurate one-step sampling.

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 $\mathbf{V}(x)=w(r(x))\,β(x)$, where $β(x)=\nabla \log (p(x)/q(x))$ 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.

Problem

• data-free sampling
• one-step sampling
• Wasserstein gradient flows
• unnormalized distributions
• f-divergence

Innovation

• Wasserstein gradient flows
• data-free one-step sampling
• f-divergence
• Log-Variance divergence
• velocity field decomposition