We study the problem of approximating the partition function of the ferromagnetic Ising model in graphs and hypergraphs. Our first result is a deterministic approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters &#x03B2; (the interaction) and &#x03BB; (the external field), except for the case |&#x03BB;|=1 (the zero-field case). A randomized algorithm (FPRAS) for all graphs, and all &#x03B2;,&#x03BB; has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the decay of correlations property. Rather, we exploit and extend machinery developed recently by Barvinok, and Patel and Regts, based on the location of the complex zeros of the partition function, which can be seen as an algorithmic realization of the classical Lee-Yang approach to phase transitions. Our approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this extension, we establish a tight version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.
Much detailed work has been done on MCMC for Ising spin configurations for several important classes of graphs, including two-dimensional lattices (e.g., @cite_44 @cite_28 @cite_11 ), random graphs and graphs of bounded degree (e.g., @cite_26 ), the complete graph (e.g., @cite_1 ) and trees (e.g., @cite_31 @cite_17 ); we do not attempt to give a comprehensive summary of this line of work here.
@cite_26 @cite_28 @cite_1 @cite_17 @cite_44 @cite_31 @cite_11
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In 1999 Kannan, Tet ali and Vempala proposed a MCMC method to uniformly sample all possible realizations of a given graphical degree sequence and conjectured its rapidly mixing nature. Recently their conjecture was proved affirmative for regular graphs (by Cooper, Dyer and Greenhill, 2007), for regular directed graphs (by Greenhill, 2011) and for half-regular bipartite graphs (by Miklos, Erdős and Soukup, 2013). Several heuristics on counting the number of possible realizations exist (via sampling processes), and while they work well in practice, so far no approximation guarantees exist for such an approach. This paper is the first to develop a method for counting realizations with provable approximation guarantee. In fact, we solve a slightly more general problem; besides the graphical degree sequence a small set of forbidden edges is also given. We show that for the general problem (which contains the Greenhill problem and the Miklos, Erdős and Soukup problem as special cases) the derived MCMC process is rapidly mixing. Further, we show that this new problem is self-reducible therefore it provides a fully polynomial randomized approximation scheme (a.k.a. FPRAS) for counting of all realizations. Recent advances in Bayesian learning with large-scale data have witnessed emergence of stochastic gradient MCMC algorithms (SG-MCMC), such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian MCMC (SGHMC), and the stochastic gradient thermostat. While finite-time convergence properties of the SGLD with a 1st-order Euler integrator have recently been studied, corresponding theory for general SG-MCMCs has not been explored. In this paper we consider general SG-MCMCs with high-order integrators, and develop theory to analyze finite-time convergence properties and their asymptotic invariant measures. Our theoretical results show faster convergence rates and more accurate invariant measures for SG-MCMCs with higher-order integrators. For example, with the proposed efficient 2nd-order symmetric splitting integrator, the mean square error (MSE) of the posterior average for the SGHMC achieves an optimal convergence rate of L-4 5 at L iterations, compared to L-2 3 for the SGHMC and SGLD with 1st-order Euler integrators. Furthermore, convergence results of decreasing-step-size SG-MCMCs are also developed, with the same convergence rates as their fixed-step-size counterparts for a specific decreasing sequence. Experiments on both synthetic and real datasets verify our theory, and show advantages of the proposed method in two large-scale real applications. We study the underdamped Langevin diffusion when the log of the target distribution is smooth and strongly concave. We present a MCMC algorithm based on its discretization and show that it achieves @math error (in 2-Wasserstein distance) in @math steps. This is a significant improvement over the best known rate for overdamped Langevin MCMC, which is @math steps under the same smoothness concavity assumptions. The underdamped Langevin MCMC scheme can be viewed as a version of Hamiltonian Monte Carlo (HMC) which has been observed to outperform overdamped Langevin MCMC methods in a number of application areas. We provide quantitative rates that support this empirical wisdom. Markov chain Monte Carlo (MCMC) methods have been used in many fields (physics, chemistry, biology, and computer science) for simulation, inference, and optimization. In many applications, Markov chains are simulated for sampling from target probabilities π(X) defined on graphs G. The graph vertices represent elements of the system, the edges represent spatial relationships, while X is a vector of variables on the vertices which often take discrete values called labels or colors. Designing efficient Markov chains is a challenging task when the variables are strongly coupled. Because of this, methods such as the single-site Gibbs sampler often experience suboptimal performance. A well-celebrated algorithm, the Swendsen–Wang (SW) method, can address the coupling problem. It clusters the vertices as connected components after turning off some edges probabilistically, and changes the color of one cluster as a whole. It is known to mix rapidly under certain conditions. Unfortunately, the SW method has limited ...