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

TL;DR

This paper introduces a unified theoretical framework for data-free one-step sampling from unnormalized distributions using Wasserstein gradient flows, revealing a universal velocity field structure across divergence objectives.

Contribution

It formalizes the structure of divergence-driven drifts, extends to Log-Variance divergence, and proposes practical implementations for efficient one-step sampling.

Findings

The velocity field has a universal form across divergence objectives.

The framework accurately predicts sampling behavior on Gaussian mixtures.

Proposed methods enable effective one-step inference after training.

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