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

• URL: http://arxiv.org/abs/2605.17808v1
• Date: Mon, 18 May 2026 03:32:22 GMT
• Title: A Unified Framework for Data-Free One-Step Sampling via Wasserstein Gradient Flows
• Abstract summary: We develop a unified theoretical framework for data-free one-step sampling from unnormalized target distributions based on Wasserstein flows.<n>For a broad class of standard f-divergence objectives, we show that the induced velocity gradient field admits the universal form $mathbfV(x)=w(r(x)),(x)$.<n>We 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
• Score: 6.695501393837496
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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