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We study analytically the equilibrium properties of the spherical hierarchical model in the presence of random fields. The expression for the critical line separating a paramagnetic from a ferromagnetic phase is derived. The critical…
We devise a deterministic algorithm to efficiently sample high-quality solutions of certain spin-glass systems that encode hard optimization problems. We employ tensor networks to represent the Gibbs distribution of all possible…
Graphical models for finite-dimensional spin glasses and real-world combinatorial optimization and satisfaction problems usually have an abundant number of short loops. The cluster variation method and its extension, the region graph…
We develop a statistical-mechanical formulation for image restoration and error-correcting codes. These problems are shown to be equivalent to the Ising spin glass with ferromagnetic bias under random external fields. We prove that the…
We introduce a Random Energy Model on a hierarchical lattice where the interaction strength between variables is a decreasing function of their mutual hierarchical distance, making it a non-mean field model. Through small coupling series…
We study the Potts spin glass model, which generalizes the Sherrington-Kirkpatrick model to the case when spins take more than two values but their interactions are counted only if the spins are equal. We obtain the analogue of the Parisi…
The exponential family of random graphs is one of the most promising class of network models. Dependence between the random edges is defined through certain finite subgraphs, analogous to the use of potential energy to provide dependence…
Is perfect error correction always worth the trouble? A framework is presented for the analysis of error detection and correction in multi-level systems of communication that takes into account degrees of freedom attended and ignored by…
We describe random generation algorithms for a large class of random combinatorial objects called Schur processes, which are sequences of random (integer) partitions subject to certain interlacing conditions. This class contains several…
We introduce a new family of energy-based probabilistic graphical models for efficient unsupervised learning. Its definition is motivated by the control of the spin-glass properties of the Ising model described by the weights of Boltzmann…
For a class of stochastic restart algorithms we address the effect of a nonzero level of randomization in maximizing the convergence rate for general energy landscapes. The resulting characterization of the optimal level of randomization is…
We analyze the free energy and the overlaps in the 2-spin spherical Sherrington Kirkpatrick spin glass model with an external field for the purpose of understanding the transition between this model and the one without an external field. We…
We investigate the statistical physics of quantum error correction in ${\rm U}(1)$ symmetry-enriched topological quantum memories. Starting from a phenomenological error model of charge-conserving noise, we study the optimal decoder…
In an earlier work, the statistical physics associated with finite--temperature decoding of code ensembles, along with the relation to their random coding error exponents, were explored in a framework that is analogous to Derrida's random…
Spin-glass systems are universal models for representing many-body phenomena in statistical physics and computer science. High quality solutions of NP-hard combinatorial optimization problems can be encoded into low energy states of…
The gauge theory of spin glasses and statistical-mechanical formulation of error-correcting codes are reviewed with an emphasis on their similarities. For the gauge theory, we explain the functional identities on dynamical autocorrelation…
We provide lower error bounds for randomized algorithms that approximate integrals of functions depending on an unrestricted or even infinite number of variables. More precisely, we consider the infinite-dimensional integration problem on…
We consider ensembles of channel codes that are partitioned into bins, and focus on analysis of exact random coding error exponents associated with optimum decoding of the index of the bin to which the transmitted codeword belongs. Two main…
We introduce and analyze free energy landscapes defined by associating to any point inside the sphere a free energy calculated on a thin spherical band around it, using many orthogonal replicas. This allows us to reinterpret, rigorously…
This paper presents a systematic study of the application of convolutional neural networks (CNNs) as an efficient and versatile tool for the analysis of critical and low-temperature phase states in spin system models. The problem of…