系统扫描动力学的熵近似张量化与最优混合
Optimal mixing of the systematic scan dynamics via approximate tensorization of entropy
- Pennsylvania State University（宾夕法尼亚州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过熵和方差的近似张量化，证明了系统扫描动力学在标准假设下具有最优混合时间，并应用于反铁磁双自旋系统和Potts模型。
AI中文摘要：
我们研究高维离散分布的系统扫描动力学的混合时间。该马尔可夫链按照固定的预定顺序依次更新坐标，与随机均匀选择坐标更新的Glauber动力学形成对比。系统扫描在实践中常受青睐，因为它表现出强大的经验性能，但其理论分析远不如Glauber动力学成熟。我们通过证明分布坐标间弱依赖的两个标准函数概念为系统扫描动力学提供强收敛保证，朝着解决这一不平衡迈出了一步。首先，我们证明在关于分布的标准边际、连通性和有界交互度假设下，熵的近似张量化蕴含每个扫描顺序的最优$O(\log n)$混合时间。其次，我们证明方差的近似张量化产生每次扫描方差泛函的常数因子收缩，这进而蕴含系统扫描动力学的自然加性和乘性可逆化的最优$O(1)$松弛时间。与我们的熵结果相比，方差界将最大交互度的依赖从指数改善为二次，并且对分布要求更弱的假设。作为我们结果的具体应用，我们在树唯一性区域中为有界度反铁磁双自旋系统建立了系统扫描动力学的最优$O(\log n)$混合，并在$\mathbb{Z}^2$中的方形盒子上为铁磁$q$态Potts模型在整个亚临界区域建立了最优$O(\log n)$混合。
英文摘要：
We study the mixing time of the systematic scan dynamics for high-dimensional discrete distributions. This Markov chain updates coordinates sequentially according to a fixed predetermined order, in contrast to the Glauber dynamics that updates coordinates selected uniformly at random. The systematic scan is often favored in practice because it exhibits strong empirical performance, but its theoretical analysis remains far less developed than that of Glauber dynamics. We take a step toward addressing this imbalance by showing that two standard functional notions of weak dependence between the coordinates of the distribution provide strong convergence guarantees for the systematic scan dynamics. First, we show that approximate tensorization of entropy implies optimal $O(\log n)$ mixing time for every scan order under standard marginal, connectivity, and bounded interaction degree assumptions about the distribution. Second, we show that approximate tensorization of variance yields a constant-factor contraction of the variance functional per scan, which in turn implies an optimal $O(1)$ relaxation time for the natural additive and multiplicative reversibilizations of the systematic scan dynamics. Compared with our entropy result, the variance bound improves the dependence on the maximum interaction degree from exponential to quadratic and requires weaker assumptions on the distribution. As concrete applications of our results, we establish optimal $O(\log n)$ mixing of the systematic scan dynamics for bounded-degree antiferromagnetic two-spin systems in the tree-uniqueness region and for the ferromagnetic $q$-state Potts model on square boxes in $\mathbb{Z}^2$ throughout its subcritical regime.
