对数凹性的保持与Wasserstein-Fisher-Rao梯度流的收敛性
Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flows
- University of Turin（都灵大学）
- Collegio Carlo Alberto（卡罗·阿尔贝托学院）
- UNSW Sydney（新南威尔士大学悉尼分校）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文证明WFR梯度流在强对数凹目标分布下保持对数凹性，并导出无需热启动的对称KL散度非渐近收敛速率，其可分解为Wasserstein与Fisher-Rao贡献，证实相关猜想。
中文摘要 AI 辅助
我们研究了Wasserstein-Fisher-Rao（WFR）梯度流的收敛性，该梯度流用于从已知归一化常数的概率分布中采样。通过将Wasserstein输运与Fisher-Rao生灭动力学相结合，WFR流平衡了探索与选择。这些流已被认为是超越Langevin动力学加速收敛的有前景机制。我们证明，对于一类满足额外曲率条件的强对数凹目标分布，WFR流保持强对数凹性，而Wasserstein流仅在高斯情形下享有此性质。利用这一结果，我们推导了对称Kullback-Leibler散度的显式非渐近收敛速率，无需当前估计中所需的热启动。特别地，我们证明收敛速率可加性分解为Wasserstein和Fisher-Rao贡献，从而在此设定下证实了近期的一个猜想。这些结果提供了精细的收敛保证，并进一步发展了用于采样和贝叶斯推断的WFR梯度流的理论基础。
英文摘要
We study the convergence of Wasserstein-Fisher-Rao (WFR) gradient flows for sampling from probability distributions known up to a normalisation constant. By combining Wasserstein transport with Fisher-Rao birth-death dynamics, WFR flows balance exploration and selection. These flows have been recognised as a promising mechanism to accelerate convergence beyond Langevin dynamics. We show that for a class of strongly log-concave target distributions satisfying additional curvature conditions, WFR flows preserve strong log-concavity, in contrast to Wasserstein flows which enjoy this property only in the Gaussian setting. Exploiting this result, we derive explicit non-asymptotic convergence rates for the symmetrised Kullback-Leibler divergence, without requiring a warm-start as required in current estimates. In particular, we show that the convergence rate decomposes additively into Wasserstein and Fisher-Rao contributions, thereby confirming a recent conjecture within this setting. These results provide refined convergence guarantees and further develop the theoretical foundations of WFR gradient flows for sampling and Bayesian inference.
