论文标题
部分可观测时空混沌系统的无模型预测
Localization Schemes: A Framework for Proving Mixing Bounds for Markov Chains
论文作者
论文摘要
Two recent and seemingly-unrelated techniques for proving mixing bounds for Markov chains are: (i) the framework of Spectral Independence, introduced by Anari, Liu and Oveis Gharan, and its numerous extensions, which have given rise to several breakthroughs in the analysis of mixing times of discrete Markov chains and (ii) the Stochastic Localization technique which has proven useful in establishing mixing and expansion bounds for both对数符号量度和离散超立方体的措施。在本文中，我们介绍了一个框架，该框架连接了两种技术的想法。我们的框架统一，简化并扩展了这两种技术。在其中心是本地化方案的概念，该方案在每种概率度量中都分配了概率度量的martingale，该概率度量随着时间的变化而定位在空间中。事实证明，每个这样的方案都对应了马尔可夫链，并且在此框架中自然出现了许多感兴趣的链。该观点为通过分析相应的本地化过程提供了为动力学得出混合边界的工具。光谱独立性和熵独立性概念的概括自然来自我们的定义，尤其是我们通过简单的Martingale参数恢复了光谱和熵独立框架中的主要定理（完全绕过需要使用高维扩张者理论）。我们通过在最近的文献中为许多混合界提供简短的（可以说）简单的证据来证明我们提出的机械的强度，包括在树上唯一的核心模型（任意程度）上，在Glauber动态的混合时间中给出了第一个$ O（n \ log n）$。
Two recent and seemingly-unrelated techniques for proving mixing bounds for Markov chains are: (i) the framework of Spectral Independence, introduced by Anari, Liu and Oveis Gharan, and its numerous extensions, which have given rise to several breakthroughs in the analysis of mixing times of discrete Markov chains and (ii) the Stochastic Localization technique which has proven useful in establishing mixing and expansion bounds for both log-concave measures and for measures on the discrete hypercube. In this paper, we introduce a framework which connects ideas from both techniques. Our framework unifies, simplifies and extends those two techniques. In its center is the concept of a localization scheme which, to every probability measure, assigns a martingale of probability measures which localize in space as time evolves. As it turns out, to every such scheme corresponds a Markov chain, and many chains of interest appear naturally in this framework. This viewpoint provides tools for deriving mixing bounds for the dynamics through the analysis of the corresponding localization process. Generalizations of concepts of Spectral Independence and Entropic Independence naturally arise from our definitions, and in particular we recover the main theorems in the spectral and entropic independence frameworks via simple martingale arguments (completely bypassing the need to use the theory of high-dimensional expanders). We demonstrate the strength of our proposed machinery by giving short and (arguably) simpler proofs to many mixing bounds in the recent literature, including giving the first $O(n \log n)$ bound for the mixing time of Glauber dynamics on the hardcore-model (of arbitrary degree) in the tree-uniqueness regime.
