A very fast inference algorithm for finite-dimensional spin glasses: Belief Propagation on the dual lattice

Starting from a Cluster Variational Method, and inspired by the correctness of the paramagnetic Ansatz (at high temperatures in general, and at any temperature in the 2D Edwards-Anderson model) we propose a novel message passing algorithm --- the Dual algorithm --- to estimate the marginal probabilities of spin glasses on finite dimensional lattices. We show that in a wide range of temperatures our algorithm compares very well with Monte Carlo simulations, with the Double Loop algorithm and with exact calculation of the ground state of 2D systems with bimodal and Gaussian interactions. Moreover it is usually 100 times faster than other provably convergent methods, as the Double Loop algorithm.Comment: 23 pages, 12 figures. v2: improved introductio

Optimal Resource Allocation in Random Networks with Transportation Bandwidths

We apply statistical physics to study the task of resource allocation in random sparse networks with limited bandwidths for the transportation of resources along the links. Useful algorithms are obtained from recursive relations. Bottlenecks emerge when the bandwidths are small, causing an increase in the fraction of idle links. For a given total bandwidth per node, the efficiency of allocation increases with the network connectivity. In the high connectivity limit, we find a phase transition at a critical bandwidth, above which clusters of balanced nodes appear, characterised by a profile of homogenized resource allocation similar to the Maxwell's construction.Comment: 28 pages, 11 figure

Inference and Optimization of Real Edges on Sparse Graphs - A Statistical Physics Perspective

Inference and optimization of real-value edge variables in sparse graphs are studied using the Bethe approximation and replica method of statistical physics. Equilibrium states of general energy functions involving a large set of real edge-variables that interact at the network nodes are obtained in various cases. When applied to the representative problem of network resource allocation, efficient distributed algorithms are also devised. Scaling properties with respect to the network connectivity and the resource availability are found, and links to probabilistic Bayesian approximation methods are established. Different cost measures are considered and algorithmic solutions in the various cases are devised and examined numerically. Simulation results are in full agreement with the theory.Comment: 21 pages, 10 figures, major changes: Sections IV to VII updated, Figs. 1 to 3 replace

A Deterministic and Generalized Framework for Unsupervised Learning with Restricted Boltzmann Machines

Restricted Boltzmann machines (RBMs) are energy-based neural-networks which are commonly used as the building blocks for deep architectures neural architectures. In this work, we derive a deterministic framework for the training, evaluation, and use of RBMs based upon the Thouless-Anderson-Palmer (TAP) mean-field approximation of widely-connected systems with weak interactions coming from spin-glass theory. While the TAP approach has been extensively studied for fully-visible binary spin systems, our construction is generalized to latent-variable models, as well as to arbitrarily distributed real-valued spin systems with bounded support. In our numerical experiments, we demonstrate the effective deterministic training of our proposed models and are able to show interesting features of unsupervised learning which could not be directly observed with sampling. Additionally, we demonstrate how to utilize our TAP-based framework for leveraging trained RBMs as joint priors in denoising problems