Interpretations of Cluster Sampling by Swendsen-Wang

Markov chain Monte Carlo (MCMC) methods have been used in many fields (physics, chemistry, biology, and computer science) for simulation, inference, and optimization. The essence of these methods is to simulate a Markov chain whose state X follows a target probability X ∼ π(X). In many applications, π(X) is defined on a graph G whose vertices represent elements in the system and whose edges represent the connectivity of the elements. X is a vector of variables on the vertices which often take discrete values called labels or colors. Designing rapid mixing Markov chain is a challenging task when the variables in the graph are strongly coupled. Methods, like the single-site Gibbs sampler, often experience long waiting time. A well-celebrated algorithm for sampling on graphs is the Swendsen-Wang (1987) (SW) method. The SW method finds a cluster of vertices as a connected component after turning off some edges probabilistically, and flips the color of the cluster as a whole. It is shown to mix rapidly under certain conditions. It has polynomial mixing time when the graph is a O(1) connectivity, i.e. the number of neighbors of each vertex is constant and does not grow with the graph size. In the literature, there are several ways for interpreting the SW-method which leads to various analyses or generalizations, including 1. A Metropolis-Hastings perspective: using auxiliary variables to propose the moves and accepting with probability 1. 2. Data augmentation: sampling a joint probability whose marginal probability is π(X). 3. Slice sampling and partial decoupling. Then we generalize SW from Potts model to arbitrary probabilities on graphs following the Metropolis-Hastings perspective, and derive a generalized Gibbs sampler.