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Physical Review E

Estimation of quenched random fields in the inverse Ising problem using a diagonal matching method

Hirohito Kiwata

Published Jun 26, 2014• 3 citations
Publisher

Abstract

We consider a method for the accurate estimation of quenched random fields in the inverse Ising problem. Approximations such as the mean-field or Bethe methods are applied to estimate quenched random coupling parameters and external fields. A diagonal matching method is introduced to ensure consistency of the diagonal part of the susceptibility, and the method yields an accurate estimation of the external fields. We introduce the diagonal matching method into the mean-field, Thouless-Anderson-Palmer, and Bethe approximations, and we investigate the effect of the diagonal matching method on the accuracy of estimation of the external fields.

DOI
10.1103/physreve.89.062135

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We consider a method for the accurate estimation of quenched random fields in the inverse Ising problem. Approximations such as the mean-field or Bethe methods are applied to estimate quenched random coupling parameters and external fields. A diagonal matching method is introduced to ensure consistency of the diagonal part of the susceptibility, and the method yields an accurate estimation of the external fields. We introduce the diagonal matching method into the mean-field, Thouless-Anderson-Palmer, and Bethe approximations, and we investigate the effect of the diagonal matching method on the accuracy of estimation of the external fields.
The inverse Ising problem consists in inferring the coupling constants of an Ising model given the correlation matrix. The fastest methods for solving this problem are based on mean-field approximations, but which one performs better in the general case is still not completely clear. In the first part of this work, I summarize the formulas for several mean- field approximations and I derive new analytical expressions for the Bethe approximation, which allow to solve the inverse Ising problem without running the Susceptibility Propagation algorithm (thus avoiding the lack of convergence). In the second part, I compare the accuracy of different mean field approximations on several models (diluted ferromagnets and spin glasses) defined on random graphs and regular lattices, showing which one is in general more effective. A simple improvement over these approximations is proposed. Also a fundamental limitation is found in using methods based on TAP and Bethe approximations in presence of an external field.
The partition function of the 2D Ising model with random nearest-neighbour coupling is expressed in the dual lattice made of square plaquettes. The dual model is solved in the mean field and in different types of Bethe-Peierls approximations, using the replica method. The application of methods of mean-field type to Ising models allows one to obtain very accurate approximations of the thermodynamic quantities. However, in the presence of quenched disorder this approach is difficult to implement. In this paper, we show that rather good results can be obtained after performing a duality transformation of the Ising model with random nearest-neighbour coupling that assumes the values Jij D1 with equal probability. The model is thus defined on a dual lattice where the spin variables are attached to the square plaquettes. The advantage is that the quadratic term of the dual Hamiltonian has constant coefficients instead of random ones. It is therefore possible to use the standard methods of the mean field to estimate the quenched free energy. Our results can be generalized to higher dimensions, although the approximations become rougher, because the number of plaquette spins over the number of interaction links increases with the dimensionality (1). In particular, we obtain an extremely good estimate of the ground- state energy of the random Ising model, by applying the Bethe-Peierls approximation where part of the short-range order is taken into account. In section 1, we introduce the dual lattice made of elementary square plaquettes. On this lattice the partition function can be expressed as a function of the inverse temperature e D 1 ln tanh.1=T/ whereT D 1 is the temperature of the original lattice. In section 2, we apply the mean-field approximation to the dual model, using the replica method. In section 3, we introduce the Bethe-Peierls approximation. This allows us to obtain a very precise estimate of the ground-state energy of the two-dimensional Ising model with random coupling. In section 4, we show that it is possible to improve the Bethe-Peierls approximation by introducing an interaction between different replicas. In section 5, the reader will find some remarks and conclusions.
We give explicit formulas of the Bethe approximation with multipoint correlations for systems with magnetic field. The obtained formulas include the closed form of the magnetization and the correlations between adjacent degrees of freedom. On the basis of our results, we propose a new iterative algorithm of the improved Bethe approximation. We confirm that the proposed technique is available for the random spin systems and indeed gives more accurate locations of the critical points. We discuss the possibility of the application of our method to the inverse Ising model by use of these equations.
We study the d-dimensional random Ising model using a suitable type of Bethe-Peierls approximation in the framework of the replica method. We take into account the correct interaction only inside replicated clusters of spins. Our ansatz is that the interaction of the borders of the clusters with the external world can be described via an effective interaction among replicas. The Bethe-Peierls model can be mapped into a single Ising model with a random Gaussian field, whose strength (related to the effective coupling between two replicas) is determined via a self-consistency equation. This allows us to obtain analytic estimates of the internal energy and of the critical temperature in d dimensions. \textcopyright{} 1996 The American Physical Society.
Abstract Using methods of statistical physics, we analyse the error of learning couplings in large Ising models from independent data (the inverse Ising problem). We concentrate on learning based on local cost functions, such as the pseudo-likelihood method for which the couplings are inferred independently for each spin. Assuming that the data are generated from a true Ising model, we compute the reconstruction error of the couplings using a combination of the replica method with the cavity approach for densely connected systems. We show that an explicit estimator based on a quadratic cost function achieves minimal reconstruction error, but requires the length of the true coupling vector as prior knowledge. A simple mean field estimator of the couplings which does not need such knowledge is asymptotically optimal, i.e. when the number of observations is much larger than the number of spins. Comparison of the theory with numerical simulations shows excellent agreement for data generated from two models with random couplings in the high temperature region: a model with independent couplings (Sherrington–Kirkpatrick model), and a model where the matrix of couplings has a Wishart distribution.
Quenched thermodynamics combines the equilibrium physics of fixed coupling configurations with a prescribed distribution of disorder. Problems related to quenched thermodynamics are generally intractable to approximate. We provide a quantum algorithm to calculate quenched disorder averages for the transverse-field Ising model, whose coupling configurations are organized into gauge classes. We show how to estimate these averages by preparing approximate Gibbs states through ancilla-assisted imaginary-time evolution. At each measurement, the ancilla record identifies the gauge class to which the energy estimate contributes. We validate our framework through an open 3×33\times33×3 lattice. A one-step circuit resolves all sixteen flux classes, and a finite-shot calculation estimates the quenched energy for independently distributed bond signs. Thermodynamic integration extends the reconstruction to a quenched free-energy estimate, while comparison with exact diagonalization identifies the finite-step approximation as the dominant source of error. This use of the measurement record provides access to the thermodynamics of frustrated spin systems with a disorder distribution chosen independently of the circuit's sampling frequencies.