We propose the variational quantum cavity method to construct a minimal energy subspace of wavevectors that are used to obtain some upper bounds for the energy cost of the low-temperature excitations. Given a trial wa...
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We propose the variational quantum cavity method to construct a minimal energy subspace of wavevectors that are used to obtain some upper bounds for the energy cost of the low-temperature excitations. Given a trial wavefunction we use the cavity method of statistical physics to estimate the Hamiltonian expectation and to find the optimal variational parameters in the subspace of wavevectors orthogonal to the lower-energy wavefunctions. To this end, we write the overlap between two wavefunctions within the Bethe approximation, which allows us to replace the global orthogonality constraint with some local constraints on the variational parameters. The method is applied to the transverse Ising model and different levels of approximations are compared with the exact numerical solutions for small systems.
We investigate different ways of generating approximate solutions to the inverse Ising problem (IIP). Our approach consists in taking as a starting point for further perturbation procedures, a Bethe mean-field solutio...
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We investigate different ways of generating approximate solutions to the inverse Ising problem (IIP). Our approach consists in taking as a starting point for further perturbation procedures, a Bethe mean-field solution obtained with a maximum spanning tree (MST) of pairwise mutual information, which we refer to as the Bethe reference point. We consider three different ways of following this idea: in the first one, we discuss a greedy procedure by which optimal links to be added starting from the Bethe reference point are selected and calibrated iteratively;the second one is based on the observation that the natural gradient can be computed analytically at the Bethe point;the last one deals with loop corrections to the Bethe point. Assuming no external field and using a dual transform we develop a dual loop joint model based on a well-chosen cycle basis. This leads us to identify a subclass of planar models, which we refer to as dual-loop-free models, having possibly many loops, but characterized by a singly connected dual factor graph, for which the partition function and the linear response can be computed exactly in respectively O(N) and O(N-2) operations, thanks to a dual weight propagation (DWP) messagepassing procedure that we set up. When restricted to this subclass of models, the inverse Ising problem being convex, becomes tractable at any temperature. Numerical experiments show that this can serve to some extent as a good approximation for models with dual loops.
Reputation systems seek to infer which members of a community can be trusted based on ratings they issue about each other. We construct a Bayesian inference model and simulate approximate estimates using belief propag...
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Reputation systems seek to infer which members of a community can be trusted based on ratings they issue about each other. We construct a Bayesian inference model and simulate approximate estimates using belief propagation (BP). The model is then mapped onto computing equilibrium properties of a spin glass in a random field and analyzed by employing the replica symmetric cavity approach. Having the fraction of positive ratings and the environment noise level as control parameters, we evaluate in different scenarios the robustness of the BP approximation and its theoretical performance in terms of estimation error. Regions of degraded performance are then explained by the convergence properties of the BP algorithm and by the emergence of a glassy phase.
We study the role played by dilution in the average behavior of a perceptron model with continuous coupling with the replica method. We analyze the stability of the replica symmetric solution as a function of the dilu...
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We study the role played by dilution in the average behavior of a perceptron model with continuous coupling with the replica method. We analyze the stability of the replica symmetric solution as a function of the dilution field for the generalization and memorization problems. Thanks to a Gardner-like stability analysis we show that at any fixed ratio a between the number of patterns M and the dimension N of the perceptron (alpha = M/N), there exists a critical dilution field h(c) above which the replica symmetric ansatz becomes unstable.
Cascade processes are responsible for many important phenomena in natural and social sciences. Simple models of irreversible dynamics on graphs, in which nodes activate depending on the state of their neighbors, have ...
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Cascade processes are responsible for many important phenomena in natural and social sciences. Simple models of irreversible dynamics on graphs, in which nodes activate depending on the state of their neighbors, have been successfully applied to describe cascades in a large variety of contexts. Over the past decades, much effort has been devoted to understanding the typical behavior of the cascades arising from initial conditions extracted at random from some given ensemble. However, the problem of optimizing the trajectory of the system, i.e. of identifying appropriate initial conditions to maximize (or minimize) the final number of active nodes, is still considered to be practically intractable, with the only exception being models that satisfy a sort of diminishing returns property called submodularity. Submodular models can be approximately solved by means of greedy strategies, but by definition they lack cooperative characteristics which are fundamental in many real systems. Here we introduce an efficient algorithm based on statistical physics for the optimization of trajectories in cascade processes on graphs. We show that for a wide class of irreversible dynamics, even in the absence of submodularity, the spread optimization problem can be solved efficiently on large networks. Analytic and algorithmic results on random graphs are complemented by the solution of the spread maximization problem on a real-world network (the Epinions consumer reviews network).
Neural networks are able to extract information from the timing of spikes. Here we provide new results on the behavior of the simplest neuronal model which is able to decode information embedded in temporal spike patt...
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Neural networks are able to extract information from the timing of spikes. Here we provide new results on the behavior of the simplest neuronal model which is able to decode information embedded in temporal spike patterns, the so-called tempotron. Using statistical physics techniques we compute the capacity for the case of sparse, time-discretized input, and 'material' discrete synapses, showing that the device saturates the information theoretic bounds with a statistics of output spikes that is consistent with the statistics of the inputs. We also derive two simple and highly efficient learning algorithms which are able to learn a number of associations which are close to the theoretical limit. The simplest versions of these algorithms correspond to distributed online protocols of interest for neuromorphic devices, and can be adapted to address the more biologically relevant continuous-time version of the classification problem, hopefully allowing the understanding of some aspects of synaptic plasticity.
We develop distributed algorithms for efficient spectrum access strategies in cognitive radio relay networks. In our setup, primary users permit secondary users access to the resource (spectrum) as long as they consen...
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We develop distributed algorithms for efficient spectrum access strategies in cognitive radio relay networks. In our setup, primary users permit secondary users access to the resource (spectrum) as long as they consent to aiding the primary users as relays in addition to transmitting their own data. Given a pool of primary and secondary users, we desire to optimize overall network utility by determining the best configuration/pairing of secondary users with primary users. This optimization can be stated in a form similar to the maximum weighted matching problem. Given such formulation, we develop an algorithm based on affinity propagation technique that is completely distributed in its structure. We demonstrate the convergence of the developed algorithm and show that it performs close to the optimal centralized scheme.
We review critically the concepts and the applications of Cayley Trees and Bethe Lattices in statistical mechanics in a tentative effort to remove widespread misuse of these simple, but yet important - and different -...
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We review critically the concepts and the applications of Cayley Trees and Bethe Lattices in statistical mechanics in a tentative effort to remove widespread misuse of these simple, but yet important - and different - ideal graphs. We illustrate, in particular, two rigorous techniques to deal with Bethe Lattices, based respectively on self-similarity and on the Kolmogorov consistency theorem, linking the latter with the Cavity and Belief Propagation methods, more known to the physics community. (C) 2012 Elsevier B.V. All rights reserved.
The very notion of social network implies that linked individuals interact repeatedly with each other. This notion allows them not only to learn successful strategies and adapt to them, but also to condition their own...
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The very notion of social network implies that linked individuals interact repeatedly with each other. This notion allows them not only to learn successful strategies and adapt to them, but also to condition their own behavior on the behavior of others, in a strategic forward looking manner. Game theory of repeated games shows that these circumstances are conducive to the emergence of collaboration in simple games of two players. We investigate the extension of this concept to the case where players are engaged in a local contribution game and show that rationality and credibility of threats identify a class of Nash equilibria-that we call "collaborative equilibria"-that have a precise interpretation in terms of subgraphs of the social network. For large network games, the number of such equilibria is exponentially large in the number of players. When incentives to defect are small, equilibria are supported by local structures where as when incentives exceed a threshold they acquire a nonlocal nature, which requires a "critical mass" of more than a given fraction of the players to collaborate. Therefore, when incentives are high, an individual deviation typically causes the collapse of collaboration across the whole system. At the same time, higher incentives to defect typically support equilibria with a higher density of collaborators. The resulting picture conforms with several results in sociology and in the experimental literature on game theory, such as the prevalence of collaboration in denser groups and in the structural hubs of sparse networks.
In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressed sensing. These algorithms are particularly interesting due ...
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In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressed sensing. These algorithms are particularly interesting due to their low complexity (linear in the signal dimension n). The asymptotic analysis predicts the fraction of unverified signal elements at each iteration l in the asymptotic regime where n -> l. The analysis is similar in nature to the well-known density evolution technique commonly used to analyze iterative decoding algorithms. To perform the analysis, a message-passing interpretation of NB-VB algorithms is provided. This interpretation lacks the extrinsic nature of standard message-passing algorithms to which density evolution is usually applied. This requires a number of nontrivial modifications in the analysis. The analysis tracks the average performance of the recovery algorithms over the ensembles of input signals and sensing matrices as a function of l. Concentration results are devised to demonstrate that the performance of the recovery algorithms applied to any choice of the input signal over any realization of the sensing matrix follows the deterministic results of the analysis closely. Simulation results are also provided which demonstrate that the proposed asymptotic analysis matches the performance of recovery algorithms for large but finite values of n. Compared to the existing technique for the analysis of NB-VB algorithms, which is based on numerically solving a large system of coupled differential equations, the proposed method is more accurate and simpler to implement.
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