论文标题
MCMC最佳缩放的DIRICHLET形式的收敛，并在大图上与依赖性目标分布
Convergence of Dirichlet Forms for MCMC Optimal Scaling with Dependent Target Distributions on Large Graphs
论文作者
论文摘要
马尔可夫链蒙特卡洛（MCMC）算法在统计，物理，机器学习等中发挥了重要作用，它们是唯一针对某些高维问题的唯一已知的通用和高效方法。随机步行大都市（RWM）算法是最古典的MCMC算法，对科学和工程学的发展和实践产生了很大的影响。通常，通过扩散过程的弱收敛结果研究了RWM算法在高维问题中的行为。在本文中，我们利用了Dirichlet形式的MOSCO收敛性来分析大图上的RWM算法，其目标分布是Gibbs度量，其中包括满足Markov属性的任何概率度量。 Dirichlet形式的抽象和强大的理论使我们能够直接自然地在无限维空间上工作，而Mosco收敛的概念使与RWM链相关的Dirichlet形式可以躺在更改希尔伯特空间上。通过最佳缩放问题，我们证明了Dirichlet形式在标准扩散方法上的令人印象深刻的优势。
Markov chain Monte Carlo (MCMC) algorithms have played a significant role in statistics, physics, machine learning and others, and they are the only known general and efficient approach for some high-dimensional problems. The random walk Metropolis (RWM) algorithm as the most classical MCMC algorithm, has had a great influence on the development and practice of science and engineering. The behavior of the RWM algorithm in high-dimensional problems is typically investigated through a weak convergence result of diffusion processes. In this paper, we utilize the Mosco convergence of Dirichlet forms in analyzing the RWM algorithm on large graphs, whose target distribution is the Gibbs measure that includes any probability measure satisfying a Markov property. The abstract and powerful theory of Dirichlet forms allows us to work directly and naturally on the infinite-dimensional space, and our notion of Mosco convergence allows Dirichlet forms associated with the RWM chains to lie on changing Hilbert spaces. Through the optimal scaling problem, we demonstrate the impressive strengths of the Dirichlet form approach over the standard diffusion approach.
