We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics.

Introduction Graphical models provide a formalism in which to express and manipulate conditional independence statements. Inference algorithms for graphical models exploit these independence statements, using them to compute conditional probabilities while avoiding brute force marginalization over the joint probability table. Many inference algorithms, in particular the clustering algorithms, make explicit their usage of conditional independence by constructing a data structure that captures the essential Markov properties underlying the graph. That is, the algorithm groups interacting variables into clusters, such that the hypergraph of clusters has Markov properties that allow simple local algorithms to be employed for inference. In the best case, in which the original graph is sparse and without long cycles, the clusters are small and inference is efficient. In the worst case, such as the case of a dense graph, the clusters are large and inference is inefficient (complexity

We review results concerning the critical behavior of spin systems at equilibrium. We consider the Ising and the general O(N)-symmetric universality class. For each of them, we review the estimates of the critical exponents, of the equation of state, of several amplitude ratios, and of the two-point function of the order parameter. We report results in three and two dimensions. We discuss the crossover phenomena that are observed in this class of systems. In particular, we review the field-theoretical and numerical studies of systems with medium-range interactions. Moreover, we consider several examples of magnetic and structural phase transitions, which are described by more complex Landau-Ginzburg-Wilson Hamiltonians, such as N-component systems with cubic anisotropy, O(N)-symmetric systems in the presence of quenched disorder, frustrated spin systems with noncollinear or canted order, and finally, a class of systems described by the tetragonal Landau-Ginzburg-Wilson Hamiltonian with three quartic couplings. The results for the tetragonal Hamiltonian are original, in particular we present the six-loop perturbative series for the β-functions and the critical exponents.

. We introduce a learning algorithm for unsupervised neural networks based on ideas from statistical mechanics. The algorithm is derived from a mean field approximation for large, layered sigmoid belief networks. We show how to (approximately) infer the statistics of these networks without resort to sampling. This is done by solving the mean field equations, which relate the statistics of each unit to those of its Markov blanket. Using these statistics as target values, the weights in the network are adapted by a local delta rule. We evaluate the strengths and weaknesses of these networks for problems in statistical pattern recognition. 1. Introduction Multilayer neural networks trained by backpropagation provide a versatile framework for statistical pattern recognition. They are popular for many reasons, including the simplicity of the learning rule and the potential for discovering hidden, distributed representations of the problem space. Nevertheless, there are many issues that are...

The Elastic Net Algorithm (ENA) for solving the Traveling Salesman Problem is analyzed applying statistical mechanics. Using some general properties of the free energy function of stochastic Hopfield Neural Networks, we argue why Simic's derivation of the ENA from a Hopfield network is incorrect. However, like the Hopfield-Lagrange method, the ENA may be considered a specific dynamic penalty method , where, in this case, the weights of the various penalty terms decrease during execution of the algorithm. This view on the ENA corresponds to the view resulting from the theory on `deformable templates', where the term stochastic penalty method seems to be most appropriate. Next, the ENA is analyzed both on the level of the energy function as well as on the level of the motion equations. It will be proven and shown experimentally, why a non-feasible solution is sometimes found. It can be caused either by a too rapid lowering of the temperature parameter (which is avoidable), or...

We present extensive Monte-Carlo spin dynamics simulations of the classical XY model in three dimensions on a simple cubic lattice with periodic boundary conditions. A recently developed efficient integration algorithm for the equations of motion is used, which allows a substantial improvement of statistics and large integration times. We find spin wave peaks in a wide range around the critical point and spin diffusion for all temperatures. At the critical point we find evidence for a violation of dynamic scaling in the sense that independent components of the dynamic structure factor S(q,ω) require different dynamic exponents in order to obtain scaling. Below the critical point we investigate the dispersion relation of the spin waves and the linewidths of S(q,ω) and find agreement with mode coupling theory. Apart from strong spin wave peaks we observe additional peaks in S(q,ω) which can be attributed to two-spin wave interactions. The overall lineshapes are also discussed and compared to mode coupling predictions. Finally, we present first results for the transport coefficient D(q,ω) of the out-of-plane magnetization component at the critical point, which is related to the thermal conductivity of 4 He near the superfluid-normal transition.

A new kind of fundamental superfield is proposed, with an Ising-like Euclidean action. Near the Planck energy it undergoes its first stage of symmetry-breaking, and the ordered phase is assumed to support specific kinds of topological defects. This picture leads to a low-energy Lagrangian which is similar to that of standard physics, but there are interesting and observable di#erences. For example, the cosmological constant vanishes, fermions have an extra coupling to gravity, the gravitational interaction of W-bosons is modified, and Higgs bosons have an unconventional equation of motion. e-mail: allen@phys.tamu.edu tel.: (409) 845-4341 fax: (409) 845-2590 International Journal of Modern Physics A, Vol. 12, No. 13 (1997) 2385-2412 CTP-TAMU-15/96 1 1 Introduction The terms "superfield" and "supersymmetry" are ordinarily used in a context which presupposes local Lorentz invariance. 1-3 It is far from clear, however, that Lorentz invariance is still valid near the Planck scale, fift...

We review and correct the classical critical exponents characterizing the transition from negative to positive black hole's heat capacity at high charge-- angular momentum. We discuss the stability properties of black holes as a thermodynamic system in equilibrium with a radiation bath (canonical ensamble) by using the Helmholtz free energy potential. We finally analytically extend the analysis to negative mass holes and study its thermodynamical stability behavior. 04.70.Dy,05.70.Jk Typeset using REVT E X Electronic Address: lousto@mail.physics.utah.edu I. INTRODUCTION The Kerr--Newman geometry (written in Boyer--Lindquist coordinates) ds 2 = \Gamma (\Delta \Gamma a 2 sin 2 `)q \Gamma2 dt 2 \Gamma 4Mra sin 2 `q \Gamma2 dtd' + q 2 \Delta dr 2 + q 2 d` 2 + h (r 2 + a 2 ) 2 \Gamma \Deltaa 2 sin 2 ` i q \Gamma2 sin 2 `d' 2 ; (1) where \Delta = r 2 \Gamma 2Mr + a 2 +Q 2 and q 2 = r 2 + a 2 cos 2 ` ; (2) represents the general s...

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Using the PVM programming environment for parallel applications, we have parallelized a simulation of the two--dimensional Ising Model on a cluster of IBM RISC System/6000 1 workstations connected by a Token Ring (16Mb/sec) and by Serial Optical Channels (220 Mb/sec) via a NSC 2 DX Router. The parallelization is done by dividing the lattice into sublattices, each sublattice being associated with one workstation. On each sublattice, a Metropolis algorithm using Multispin Coding techniques is used to generate new configurations. We provide numerical results concerning the number of spin updates per second, speedups, and efficiencies for various numbers of processors and lattice sizes. Keywords. Statistical Physics; Ising Model; Workstation Cluster; Geometric Parallelization. 1 Introduction The goal of Theoretical Statistical Physics is a mathematical description of thermodynamic properties (e.g. of magnetism or phase transitions) of macroscopic bodies, commercing with a description ...