Ising and Potts models are an important class of discrete probability distributions which originated from statistical physics and since then have found applications in several disciplines. Simulation from these models is a well known challenging problem. In this paper, we study a class of Markov chain Monte Carlo algorithms, in which we introduce an auxiliary Gaussian variable such that, conditional on this variable, the discrete states are independent. This approach is broadly applicable to Ising and Potts models, including ones in which the coupling matrix admits negative entries, as in spin glass and Hopfield models. We focus on a block Gibbs sampler version of this algorithm, which alternates between sampling the auxiliary Gaussian and the discrete states, and derive mixing time bounds for a wide class of Ising/Potts models at both high and low temperatures, yielding results analogous to those derived for the Heat Bath and Swendsen-Wang algorithms. We present novel choices of auxiliary Gaussian variables which scale well with the number of states in the Potts model, and which can take advantage of the low rank structure of the coupling matrix, if any. Finally, we numerically evaluate the performance of the auxiliary Gaussian Gibbs sampler with several competing algorithms, across a range of examples.

翻译：伊辛模型和波茨模型是一类重要的离散概率分布，其起源于统计物理学，并已在多个学科领域得到应用。从这些模型中进行仿真是一个公认的难题。本文研究了一类马尔可夫链蒙特卡洛算法，该类算法引入一个辅助高斯变量，使得在该变量给定的条件下，离散状态相互独立。该方法广泛适用于伊辛模型和波茨模型，包括耦合矩阵含有负元素的模型（如自旋玻璃模型和霍普菲尔德模型）。我们重点研究该算法的块吉布斯采样器版本，该版本交替采样辅助高斯变量和离散状态，并在高温和低温条件下推导了广泛伊辛/波茨模型的混合时间界，所得结果与热浴算法和Swendsen-Wang算法的结论类似。我们提出了一些新颖的辅助高斯变量选择方法，这些变量能够很好地随波茨模型的状态数量扩展，并且能够利用耦合矩阵的低秩结构（如果存在）。最后，我们通过一系列实例，将辅助高斯吉布斯采样器的性能与若干竞争算法进行了数值评估。