AI 中文总结
本文针对硬核模型开发非线性动力学分析的基础理论工具，证明两类非线性动力学在低密度下的收敛性，设计了基于粒子系统动力学的简单采样算法。
AI 中文摘要
近年来，源自动力学理论的非线性动力学在伊辛模型等自旋系统的构型采样领域受到关注。本文聚焦硬核模型的非线性动力学，硬核模型是带有硬约束的典型自旋系统，它以图的独立集大小为权重，指定了独立集上的分布。我们研究两类不同的非线性动力学：一类是平均场动力学，它保留独立集的密度（或平均大小）；另一类是单站点动力学，它保留边际向量（即顶点的占用概率）。这些动力学是从具有指定密度或边际向量的硬核模型中采样的自然随机过程，二者均是最大熵分布的典型实例，已在多种场景中得到研究。与线性马尔可夫链不同，目前缺乏针对非线性动力学的基础理论框架。本文为硬核模型背景下的非线性动力学分析开发了基础理论工具，通过新颖的耦合论证，证明了平均场动力学和单站点动力学在密度足够低时几乎线性收敛；还证明了平均场动力学的相对熵在达到临界密度前均呈指数衰减。此外，我们设计了新的算法，用于从具有指定密度或指定边际向量的硬核分布中采样，这些算法基于相关的线性马尔可夫链（称为粒子系统动力学），灵感源自所谓的Kac纲领，该链可近似相关的非线性动力学。正如本文所展示的，这些算法的时间复杂度与基于参数学习的传统方法相当，但实现更简单。
英文摘要
In recent years, nonlinear dynamics derived from kinetic theory have gained attention in the context of sampling configurations of spin systems such as the Ising model. We focus on nonlinear dynamics for the hard-core model, a canonical spin system with hard constraints that specifies a distribution over independent sets in a graph, weighted by their sizes. We explore two distinct types of nonlinear dynamics: the mean-field dynamics, which preserves the density (or average size) of independent sets, and the single-site dynamics, which preserves the marginal vector (i.e., the occupancy probabilities of the vertices). These dynamics are natural stochastic processes for sampling from the hard-core model with a specified density or marginal vector, respectively, both of which are canonical instances of maximum entropy distributions that have been studied in various contexts. In contrast to linear Markov chains, there is a significant lack of a fundamental theoretical framework for nonlinear dynamics. We develop foundational theoretical tools for analyzing nonlinear dynamics within the context of the hard-core model. We establish almost linear convergence of both the mean-field and single-site dynamics at sufficiently low density through novel coupling arguments. We also establish exponential decay of relative entropy for the mean-field dynamics all the way up to the critical density. Additionally, we design new algorithms for sampling from the hard-core distribution with either a specified density or a specified marginal vector. These algorithms are based on a related linear Markov chain, called the particle-system dynamics and inspired by the so-called Kac's program, that approximates the associated nonlinear dynamics. As we demonstrate in the paper, they are comparable in time complexity, but simpler to implement, than traditional approaches based on learning parameter values.
