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41
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.
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.
Central limit theorem for fluctuations in the high temperature region of the SherringtonKirkpatrick spin glass model
, 2008
"... ..."
Associative Data Storage and Retrieval in Neural Networks
, 1995
"... Associative storage and retrieval of binary random patterns in various neural net models with onestep thresholddetection retrieval and local learning rules are the subject of this paper. For different heteroassociation and autoassociation memory tasks, specified by the properties of the pattern s ..."
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Cited by 14 (6 self)
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Associative storage and retrieval of binary random patterns in various neural net models with onestep thresholddetection retrieval and local learning rules are the subject of this paper. For different heteroassociation and autoassociation memory tasks, specified by the properties of the pattern sets to be stored and upper bounds on the retrieval errors, we compare the performance of various models of finite as well as asymptotically infinite size. In infinite models, we consider the case of asymptotically sparse patterns, where the mean activity in a pattern vanishes, and study two asymptotic fidelity requirements: constant error probabilities and vanishing error probabilities. A signaltonoise ratio analysis is carried out for one retrieval step where the calculations are comparatively straightforward and easy. As performance measures we propose and evaluate information capacities in bits/synapse which also take into account the important property of fault tolerance. For autoasso...
Vector precoding for wireless MIMO systems and its replica analysis
 IEEE J. Sel. Areas Commun
"... We apply the replica method to analyze vector precoding, a method to reduce transmit power in antenna array communications. The analysis applies to a very general class of channel matrices. The statistics of the channel matrix enter the transmitted energy per symbol via its Rtransform. We find that ..."
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Cited by 12 (4 self)
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We apply the replica method to analyze vector precoding, a method to reduce transmit power in antenna array communications. The analysis applies to a very general class of channel matrices. The statistics of the channel matrix enter the transmitted energy per symbol via its Rtransform. We find that vector precoding performs much better for complex than for real alphabets. As a byproduct, we find a nonlinear precoding method with polynomial complexity that outperforms NPhard TomlinsonHarashima precoding for binary modulation on complex channels if the number of transmit antennas is slightly larger than twice the number of receive antennas. Index Terms Multipleantenna wireless, multipleinput multipleoutput (MIMO), spatial equalization, TomlinsonHarashima precoding, replica method, random matrices, Rtransform. I.
Replica symmetry breaking in mean field spin glasses trough HamiltonJacobi technique
, 2010
"... During the last years, through the combined effort of the insight, coming from physical intuition and computer simulation, and the exploitation of rigorous mathematical methods, the main features of the mean field SherringtonKirkpatrick spin glass model have been firmly established. In particular, ..."
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Cited by 10 (9 self)
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During the last years, through the combined effort of the insight, coming from physical intuition and computer simulation, and the exploitation of rigorous mathematical methods, the main features of the mean field SherringtonKirkpatrick spin glass model have been firmly established. In particular, it has been possible to prove the existence and uniqueness of the infinite volume limit for the free energy, and its Parisi expression, in terms of a variational principle, involving a functional order parameter. Even the expected property of ultrametricity, for the infinite volume states, seems to be near to a complete proof. The main structural feature of this model, and related models, is the deep phenomenon of spontaneous replica symmetry breaking (RSB), discovered by Parisi many years ago. By expanding on our previous work, the aim of this paper is to investigate a general frame, where replica symmetry breaking is embedded in a kind of mechanical scheme of the HamiltonJacobi type. Here, the analog of the “time ” variable is a parameter characterizing the strength of the interaction, while the “space” variables rule out quantitatively the broken replica symmetry pattern. Starting from the simple cases, where annealing is assumed, or replica symmetry, we build up a progression of dynamical systems, with an increasing number of space variables, which allow to weaken the effect of the potential in the HamiltonJacobi equation, as the level of symmetry braking is increased. This new machinery allows to work out mechanically the general Kstep RSB solutions, in a different interpretation with respect to the replica trick, and lightens easily their properties as existence or uniqueness.
Mathematical Aspects of mean field spin glass theory
 European Congress of Mathematical Physics, ArXiv:condmat/0410435
"... A comprehensive review will be given about the rich mathematical structure of mean field spin glass theory, mostly developed, until now, in the frame of the methods of theoretical physics, based on deep physical intuition and hints coming from numerical simulation. Central to our treatment is a very ..."
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Cited by 8 (1 self)
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A comprehensive review will be given about the rich mathematical structure of mean field spin glass theory, mostly developed, until now, in the frame of the methods of theoretical physics, based on deep physical intuition and hints coming from numerical simulation. Central to our treatment is a very simple and yet powerful interpolation method, allowing to compare different probabilistic schemes, by using convexity and positivity arguments. In this way we can prove the existence of the thermodynamic limit for the free energy density of the system, a long standing open problem. Moreover, in the frame of a generalized variational principle, we can show the emergency of the DerridaRuelle random probability cascades, leading to the form of free energy given by the celebrated Parisi Ansatz. All these results seem to be in full agreement with the mechanism of spontaneous replica symmetry breaking as developed by Giorgio Parisi.
Applications of Large Random Matrices in Communications Engineering
"... This work gives an overview of analytic tools to the design, analysis, and modelling of communication systems which can be described by linear vector channels such as y = Hx+z where the number of components in each vector is large. Tools from probability theory, operator algebra, and statistical phy ..."
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Cited by 7 (3 self)
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This work gives an overview of analytic tools to the design, analysis, and modelling of communication systems which can be described by linear vector channels such as y = Hx+z where the number of components in each vector is large. Tools from probability theory, operator algebra, and statistical physics are reviewed. The survey of analytical tools is complemented by examples of applications in communications engineering.
Spin glasses
, 2005
"... From a physical point of view, spin glasses, as dilute magnetic alloys, are very interesting systems. They are characterized by such features as exhibiting a new magnetic phase, where magnetic moments are frozen into disordered ..."
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Cited by 6 (1 self)
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From a physical point of view, spin glasses, as dilute magnetic alloys, are very interesting systems. They are characterized by such features as exhibiting a new magnetic phase, where magnetic moments are frozen into disordered