加权拉普拉斯流：一种具有可证收敛性的确定性粒子流方法

中文摘要 AI 辅助

从目标概率密度中采样是统计学、机器学习与科学计算领域的基础任务。本文提出加权拉普拉斯流（weighted Laplacian flow），这是一种确定性粒子流方法，可将来自易处理初始密度的样本传输至仅需归一化的目标密度。该方法演化目标密度与传输后分布间的对数密度比，通过求解与目标密度相关的加权泊松方程构造粒子速度，此设计无需选择核函数，且可沿流直接控制粒子权重。我们建立了所提偏微分方程系统的全局适定性，证明传输后密度在$L^\finf$距离与Kullback-Leibler散度下均收敛至目标密度；在次线性强迫条件下，该方法可在有限时间内实现精确收敛。对多模态、重尾及十维目标的数值实验表明，加权拉普拉斯流可实现长程质量传输，克服能量势垒。

英文摘要

We introduce weighted Laplacian flow (WLF), a deterministic particle-flow framework for sampling from a target distribution known only up to normalization. The proposed method evolves the logarithmic density ratio through a transport equation and determines the particle velocity from a target-weighted Poisson problem, resulting in a nonlocal and kernel-free mechanism for redistributing mass. For bounded smooth domains, we establish global well-posedness of the flow for Lipschitz initial data and prove that the flow contracts the discrepancy between the evolving and target densities at an exact exponential rate in terms of the oscillation of the log-density ratio, as well as in the associated projective metric. The relative entropy also satisfies a precise dissipation relation involving the two directional Kullback--Leibler divergences. In addition, we identify a variational interpretation of WLF. The method can be viewed as a gradient flow of the reverse Kullback--Leibler divergence under a target-anchored metric. Consequently, the method achieves an explicit relaxation scale without requiring log-concavity or spectral-gap assumptions on the target distribution. Numerical experiments on multimodal, heavy-tailed, and high-dimensional targets demonstrate the effectiveness of the proposed approach in capturing long-range mass transport.

发表机构

• Tsinghua University（清华大学）
• The University of Hong Kong（香港大学）
• Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
• Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）

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