通过Dobrushin收缩的局部几何混合及其在扩散路径蒙特卡洛和近端采样器中的应用
Local Geometric Mixing via Dobrushin Contraction with Applications to Diffusion Path Monte Carlo and the Proximal Sampler
- Institute for Applied Mathematics（应用数学研究所）
- University of Bonn（波恩大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过Dobrushin收缩建立局部几何混合界，并将其应用于扩散路径蒙特卡洛及近端采样器，在最少假设下提供混合保证，并将谱间隙估计发展为混合时间界。
AI中文摘要：
局部几何混合通过仅在有限次转移内要求全变差距离下几何收敛到平衡态，从而将几何混合局部化。它适应局部收敛速率，并能捕捉快速的局部平衡过程，即使全局混合要慢得多。我们通过Dobrushin收缩建立并讨论了局部几何混合界。随后，我们将该方法应用于扩散路径蒙特卡洛（Diffusion Path Monte Carlo），这是一种近期提出的马尔可夫链蒙特卡洛方法，旨在利用基于分数的建模的进展，其理想转移与近端采样器（Proximal Sampler）的转移一致。我们的分析涵盖了理想方法及其可实现的Metropolis调整版本，在最少的假设下提供了混合保证。对于理想方法，这些保证补充了最近的谱间隙估计，我们将这些估计发展为混合时间界。
英文摘要：
Local geometric mixing localizes geometric mixing by requiring geometric convergence to equilibrium in total variation only over finitely many transitions. It accommodates local convergence rates and captures rapid local equilibration, even when global mixing is much slower. We establish and discuss local geometric mixing bounds through Dobrushin contraction. We then apply this approach to Diffusion Path Monte Carlo, a recently proposed Markov chain Monte Carlo method, aimed at leveraging advances in score-based modeling, whose ideal transitions coincide with those of the Proximal Sampler. Our analysis covers both the ideal method and its implementable Metropolis-adjusted counterpart, providing mixing guarantees under minimal assumptions. For the ideal method, these guarantees complement recent spectral gap estimates, which we develop into mixing time bounds.
