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84
Constructing Free Energy Approximations and Generalized Belief Propagation Algorithms
 IEEE Transactions on Information Theory
, 2005
"... Important inference problems in statistical physics, computer vision, errorcorrecting coding theory, and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs. The belief propagation (BP) algorithm is an efficient way to solve these problems t ..."
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Cited by 585 (13 self)
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Important inference problems in statistical physics, computer vision, errorcorrecting coding theory, and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs. The belief propagation (BP) algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree, but only approximate when the factor graph has cycles. We show that BP fixed points correspond to the stationary points of the Bethe approximation of the free energy for a factor graph. We explain how to obtain regionbased free energy approximations that improve the Bethe approximation, and corresponding generalized belief propagation (GBP) algorithms. We emphasize the conditions a free energy approximation must satisfy in order to be a “valid ” or “maxentnormal ” approximation. We describe the relationship between four different methods that can be used to generate valid approximations: the “Bethe method, ” the “junction graph method, ” the “cluster variation method, ” and the “region graph method.” Finally, we explain how to tell whether a regionbased approximation, and its corresponding GBP algorithm, is likely to be accurate, and describe empirical results showing that GBP can significantly outperform BP.
Broken replica symmetry bounds in the mean field spin glass model
 Comm. Math Phys
, 2003
"... By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the SherringtonKirkpatrick model, and the Derrida pspin model. Here we extend this argument in order to compare the limiting free energy w ..."
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Cited by 146 (15 self)
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By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the SherringtonKirkpatrick model, and the Derrida pspin model. Here we extend this argument in order to compare the limiting free energy with the expression given by the Parisi Ansatz, and including full spontaneous replica symmetry breaking. Our main result is that the quenched average of the free energy is bounded from below by the value given in the Parisi Ansatz, uniformly in the size of the system. Moreover, the difference between the two expressions is given in the form of a sum rule, extending our previous work on the comparison between the true free energy and its replica symmetric SherringtonKirkpatrick approximation. We give also a variational bound for the infinite volume limit of the ground state energy per site.
The Parisi formula
, 2006
"... Using Guerra’s interpolation scheme, we compute the free energy of the SherringtonKirkpatrick model for spin glasses at any temperature, confirming a celebrated prediction of G. Parisi. ..."
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Cited by 130 (4 self)
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Using Guerra’s interpolation scheme, we compute the free energy of the SherringtonKirkpatrick model for spin glasses at any temperature, confirming a celebrated prediction of G. Parisi.
The Thermodynamic Limit in Mean Field Spin Glass Models
 Commun. Math. Phys
, 2002
"... We present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities, for a large class of mean field disordered models, as for example the SherringtonKirkpatrick model, and the Derrida pspin model. The main argument is based on a smoo ..."
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Cited by 116 (22 self)
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We present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities, for a large class of mean field disordered models, as for example the SherringtonKirkpatrick model, and the Derrida pspin model. The main argument is based on a smooth interpolation between a large system, made of N spin sites, and two similar but independent subsystems, made of N1 and N2 sites, respectively, with N1 + N2 = N. The quenched average of the free energy turns out to be subadditive with respect to the size of the system. This gives immediately convergence of the free energy per site, in the infinite volume limit. Moreover, a simple argument, based on concentration of measure, gives the almost sure convergence, with respect to the external noise. Similar results hold also for the ground state energy per site.
About The Overlap Distribution In Mean Field Spin Glass Models
 Int. J. Phys. B
, 1997
"... . We continue our presentation of mathematically rigorous results about the SherringtonKirkpatrick mean field spin glass model. Here we establish some properties of the distribution of overlaps between real replicas. They are in full agreement with the Parisi accepted picture of spontaneous replica ..."
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Cited by 42 (11 self)
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. We continue our presentation of mathematically rigorous results about the SherringtonKirkpatrick mean field spin glass model. Here we establish some properties of the distribution of overlaps between real replicas. They are in full agreement with the Parisi accepted picture of spontaneous replica symmetry breaking. As a byproduct, we show that the selfaveraging of the EdwardsAnderson fluctuating order parameter, with respect to the external quenched noise, implies that the overlap distribution is given by the SherringtonKirkpatrick replica symmetric Ansatz. This extends previous results of Pastur and Shcherbina. Finally, we show how to generalize our results to realistic short range spin glass models. Dedicated to the memory of Hiroomi Umezawa z Research supported in part by MURST (Italian Minister of University and Scientific and Technological Research) and INFN (Italian National Institute for Nuclear Physics). 1. INTRODUCTION. The physical content of the SherringtonKirkpatr...
Quadratic replica coupling in the SherringtonKirkpatrick mean field spin glass model
, 2008
"... ..."
The infinite volume limit in generalized mean field disordered models
 Markov Processes and Related Fields 9
, 2003
"... We generalize the strategy, we recently introduced to prove the existence of the thermodynamic limit for the SherringtonKirkpatrick and pspin models, to a wider class of mean field spin glass systems, including models with multicomponent and nonIsing type spins, mean field spin glasses with an a ..."
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Cited by 26 (2 self)
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We generalize the strategy, we recently introduced to prove the existence of the thermodynamic limit for the SherringtonKirkpatrick and pspin models, to a wider class of mean field spin glass systems, including models with multicomponent and nonIsing type spins, mean field spin glasses with an additional CurieWeiss interaction, and systems consisting of several replicas of the spin glass model, where replicas are coupled with terms depending on the mutual overlaps.
Computational complexity of the landscape
 I
"... Abstract: We study the computational complexity of the physical problem of finding vacua of string theory which agree with data, such as the cosmological constant, and show that such problems are typically NP hard. In particular, we prove that in the BoussoPolchinski model, the problem is NP comple ..."
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Cited by 25 (2 self)
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Abstract: We study the computational complexity of the physical problem of finding vacua of string theory which agree with data, such as the cosmological constant, and show that such problems are typically NP hard. In particular, we prove that in the BoussoPolchinski model, the problem is NP complete. We discuss the issues this raises and the possibility that, even if we were to find compelling evidence that some vacuum of string theory describes our universe, we might never be able to find that vacuum explicitly. In a companion paper, we apply this point of view to the question of how early cosmology might select a vacuum. Contents
From neuron to neural network dynamics
, 2006
"... This paper presents an overview of some techniques and concepts coming from dynamical system theory and used for the analysis of dynamical neural networks models. In a first section, we describe the dynamics of the neuron, starting from the HodgkinHuxley description, which is somehow the canonical ..."
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Cited by 21 (8 self)
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This paper presents an overview of some techniques and concepts coming from dynamical system theory and used for the analysis of dynamical neural networks models. In a first section, we describe the dynamics of the neuron, starting from the HodgkinHuxley description, which is somehow the canonical description for the “biological neuron”. We discuss some models reducing
The stochastic traveling salesman problem: Finite size scaling and the cavity prediction
 A.G. Percus and D.C. Torney, “Greedy algorithms for optimized DNA sequencing”, Proceedings of the 10th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 1999
"... We study the random link traveling salesman problem, where lengths l ij between city i and city j are taken to be independent, identically distributed random variables. We discuss a theoretical approach, the cavity method, that has been proposed for finding the optimum tour length over this random e ..."
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Cited by 17 (3 self)
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We study the random link traveling salesman problem, where lengths l ij between city i and city j are taken to be independent, identically distributed random variables. We discuss a theoretical approach, the cavity method, that has been proposed for finding the optimum tour length over this random ensemble, given the assumption of replica symmetry. Using finite size scaling and a renormalized model, we test the cavity predictions against the results of simulations, and find excellent agreement over a range of distributions. We thus provide numerical evidence that the replica symmetric solution to this problem is the correct one. Finally, we note a surprising result concerning the distribution of kthnearest neighbor links in optimal tours, and invite a theoretical understanding of this phenomenon.