arXiv 2111.040662021-11cs.DS

谱独立性超越有界度图的快速采样

Fast sampling via spectral independence beyond bounded-degree graphs

Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, Daniel Štefankovič

备注：: TALG, To Appear

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AI总结：

本文通过摊销递推子树递归捕获L^p分析，放宽谱独立性的有界度假设，推广至随机图G(n,d/n)，提出适用于所有d的n^{1+o(1)}快速采样算法。

摘要：

谱独立性是一个近期发展的框架，用于获得经典格劳伯动力学收敛时间的尖锐界限。这一新框架已在有界度图上，对于整个所谓的唯一性区域内的一大类问题，产生了最优的$O(n \log n)$采样算法，包括例如独立集、匹配和伊辛模型配置的采样问题。我们的主要贡献是放宽了迄今为止在建立和应用谱独立性方面重要的有界度假设。先前避免度界限的方法依赖于使用$L^p$-范数来分析具有有界连通常数的图上的收缩（Sinclair, Srivastava, Yin; FOCS'13）。$L^p$-范数的非线性是应用这些结果来界定谱独立性的障碍。我们的解决方案是通过在用于分析收缩的递推的子树间进行摊销，递归地捕获$L^p$-分析。我们的方法推广了先前仅适用于有界度图的分析。作为我们技术的一个主要应用，我们考虑随机图$G(n,d/n)$，其中先前已知的算法运行时间为$n^{O(\log d)}$或仅适用于大的$d$。我们显著改进了这些算法界限，并开发了基于格劳伯动力学的快速$n^{1+o(1)}$算法，这些算法适用于所有$d$，在整个唯一性区域内。

英文摘要：

Spectral independence is a recently-developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal $O(n \log n)$ sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using $L^p$-norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, Yin; FOCS'13). The non-linearity of $L^p$-norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the $L^p$-analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph $G(n,d/n)$, where the previously known algorithms run in time $n^{O(\log d)}$ or applied only to large $d$. We refine these algorithmic bounds significantly, and develop fast $n^{1+o(1)}$ algorithms based on Glauber dynamics that apply to all $d$, throughout the uniqueness regime.

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摘要：

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