arXiv subjects

Federico Ricci-Tersenghi

Publications and source records attributed to Federico Ricci-Tersenghi.

At least 91 records · Page 5Linked to original sources

arXiv2016

Gauge-free cluster variational method by maximal messages and moment matching

We present a new implementation of the Cluster Variational Method (CVM) as a message passing algorithm. The kind of message passing algorithms used for CVM, usually named Generalized Belief Propagation, are a generalization of the Belief Propagation algorithm in the same way that CVM is a generalization of the Bethe approximation for estimating the partition function. However, the connection between fixed points of GBP and the extremal points of the CVM free-energy is usually not a one-to-one correspondence, because of the existence of a gauge transformation involving the GBP messages. Our contribution is twofold. Firstly we propose a new way of defining messages (fields) in a generic CVM approximation, such that messages arrive on a given region from all its ancestors, and not only from its direct parents, as in the standard Parent-to-Child GBP. We call this approach maximal messages. Secondly we focus on the case of binary variables, re-interpreting the messages as fields enforcing the consistency between the moments of the local (marginal) probability distributions. We provide a precise rule to enforce all consistencies, avoiding any redundancy, that would otherwise lead to a gauge transformation on the messages. This moment matching method is gauge free, i.e. it guarantees that the resulting GBP is not gauge invariant. We apply our maximal messages and moment matching GBP to obtain an analytical expression for the critical temperature of the Ising model in general dimensions at the level of plaquette-CVM. The values obtained outperform Bethe estimates, and are comparable with loop corrected Belief Propagation equations. The method allows for a straightforward generalization to disordered systems.

cond-mat.dis-nn↗

arXiv2016

Improving variational methods via pairwise linear response identities

Inference methods are often formulated as variational approximations: these approximations allow easy evaluation of statistics by marginalization or linear response, but these estimates can be inconsistent. We show that by introducing constraints on covariance, one can ensure consistency of linear response with the variational parameters, and in so doing inference of marginal probability distributions is improved. For the Bethe approximation and its generalizations, improvements are achieved with simple choices of the constraints. The approximations are presented as variational frameworks; iterative procedures related to message passing are provided for finding the minima.

stat.ML↗

arXiv2016

The backtracking survey propagation algorithm for solving random K-SAT problems

Discrete combinatorial optimization has a central role in many scientific disciplines, however, for hard problems we lack linear time algorithms that would allow us to solve very large instances. Moreover, it is still unclear what are the key features that make a discrete combinatorial optimization problem hard to solve. Here we study random K-satisfiability problems with $K=3,4$, which are known to be very hard close to the SAT-UNSAT threshold, where problems stop having solutions. We show that the backtracking survey propagation algorithm, in a time practically linear in the problem size, is able to find solutions very close to the threshold, in a region unreachable by any other algorithm. All solutions found have no frozen variables, thus supporting the conjecture that only unfrozen solutions can be found in linear time, and that a problem becomes impossible to solve in linear time when all solutions contain frozen variables.

cs.CC↗

arXiv2016

Data quality for the inverse Ising problem

There are many methods proposed for inferring parameters of the Ising model from given data, that is a set of configurations generated according to the model itself. However little attention has been paid until now to the data, e.g. how the data is generated, whether the inference error using one set of data could be smaller than using another set of data, etc. In this paper we address the data quality problem in the kinetic inverse Ising problem. We quantify the quality of data using effective rank of the correlation matrix, and show that data gathered in a out of-equilibrium regime has a better quality than data gathered in equilibrium for coupling reconstruction. We also propose a matrix-perturbation based method for tuning the quality of given data and for removing bad-quality (i.e. redundant) configurations from data.

cond-mat.stat-mech↗

arXiv2016

Solving the inverse Ising problem by mean-field methods in a clustered phase space with many states

In this work we explain how to properly use mean-field methods to solve the inverse Ising problem when the phase space is clustered, that is many states are present. The clustering of the phase space can occur for many reasons, e.g. when a system undergoes a phase transition. Mean-field methods for the inverse Ising problem are typically used without taking into account the eventual clustered structure of the input configurations and may led to very bad inference (for instance in the low temperature phase of the Curie-Weiss model). In the present work we explain how to modify mean-field approaches when the phase space is clustered and we illustrate the effectiveness of the new method on different clustered structures (low temperature phases of Curie-Weiss and Hopfield models).

cond-mat.dis-nn↗

arXiv2016

Performance of a community detection algorithm based on semidefinite programming

The problem of detecting communities in a graph is maybe one the most studied inference problems, given its simplicity and widespread diffusion among several disciplines. A very common benchmark for this problem is the stochastic block model or planted partition problem, where a phase transition takes place in the detection of the planted partition by changing the signal-to-noise ratio. Optimal algorithms for the detection exist which are based on spectral methods, but we show these are extremely sensible to slight modification in the generative model. Recently Javanmard, Montanari and Ricci-Tersenghi (arXiv:1511.08769) have used statistical physics arguments, and numerical simulations to show that finding communities in the stochastic block model via semidefinite programming is quasi optimal. Further, the resulting semidefinite relaxation can be solved efficiently, and is very robust with respect to changes in the generative model. In this paper we study in detail several practical aspects of this new algorithm based on semidefinite programming for the detection of the planted partition. The algorithm turns out to be very fast, allowing the solution of problems with $O(10^5)$ variables in few second on a laptop computer.

stat.ML↗

arXiv2016

Phase Transitions in Semidefinite Relaxations

Statistical inference problems arising within signal processing, data mining, and machine learning naturally give rise to hard combinatorial optimization problems. These problems become intractable when the dimensionality of the data is large, as is often the case for modern datasets. A popular idea is to construct convex relaxations of these combinatorial problems, which can be solved efficiently for large scale datasets. Semidefinite programming (SDP) relaxations are among the most powerful methods in this family, and are surprisingly well-suited for a broad range of problems where data take the form of matrices or graphs. It has been observed several times that, when the `statistical noise' is small enough, SDP relaxations correctly detect the underlying combinatorial structures. In this paper we develop asymptotic predictions for several `detection thresholds,' as well as for the estimation error above these thresholds. We study some classical SDP relaxations for statistical problems motivated by graph synchronization and community detection in networks. We map these optimization problems to statistical mechanics models with vector spins, and use non-rigorous techniques from statistical mechanics to characterize the corresponding phase transitions. Our results clarify the effectiveness of SDP relaxations in solving high-dimensional statistical problems.

cond-mat.stat-mech↗

arXiv2015

Quasi equilibrium construction for the long time limit of glassy dynamics

In this paper we review a recent proposal to understand the long time limit of glassy dynamics in terms of an appropriate Markov Chain. [1]. The advantages of the resulting construction are many. The first one is that it gives a quasi equilibrium description on how glassy systems explore the phase space in the slow relaxation part of their dynamics. The second one is that it gives an alternative way to obtain dynamical equations starting from a dynamical rule that is static in spirit. This provides a way to overcome the difficulties encountered in the short time part of the dynamics where current conservation must be enforced. We study this approach in detail in a prototypical mean field disordered spin system, namely the p-spin spherical model, showing how we can obtain the well known equations that describes its dynamics. Then we apply the same approach to structural glasses. We first derive a set of dynamical Ornstein-Zernike equations which are very general in nature. Finally we consider two possible closure schemes for them, namely the Hypernetted Chain approximation of liquid theory and a closure of the BBGKY hierarchy that has been recently introduced by G. Szamel. From both approaches we finally find a set of dynamical Mode-Coupling like equations that are supposed to describe the system in the long time/slow dynamics regime.

cond-mat.dis-nn↗

arXiv2015

Multiple phases in modularity-based community detection

Detecting communities in a network, based only on the adjacency matrix, is a problem of interest to several scientific disciplines. Recently, Zhang and Moore have introduced an algorithm in [P. Zhang and C. Moore, Proceedings of the National Academy of Sciences 111, 18144 (2014)], called mod-bp, that avoids overfitting the data by optimizing a weighted average of modularity (a popular goodness-of-fit measure in community detection) and entropy (i.e. number of configurations with a given modularity). The adjustment of the relative weight, the "temperature" of the model, is crucial for getting a correct result from mod-bp. In this work we study the many phase transitions that mod-bp may undergo by changing the two parameters of the algorithm: the temperature $T$ and the maximum number of groups $q$. We introduce a new set of order parameters that allow to determine the actual number of groups $\hat{q}$, and we observe on both synthetic and real networks the existence of phases with any $\hat{q} \in \{1,q\}$, which were unknown before. We discuss how to interpret the results of mod-bp and how to make the optimal choice for the problem of detecting significant communities.

physics.soc-ph↗

arXiv2015

Egalitarianism in the rank aggregation problem: a new dimension for democracy

Winner selection by majority, in an election between two candidates, is the only rule compatible with democratic principles. Instead, when the candidates are three or more and the voters rank candidates in order of preference, there are no univocal criteria for the selection of the winning (consensus) ranking and the outcome is known to depend sensibly on the adopted rule. Building upon XVIII century Condorcet theory, whose idea was to maximize total voter satisfaction, we propose here the addition of a new basic principle (dimension) to guide the selection: satisfaction should be distributed among voters as equally as possible. With this new criterion we identify an optimal set of rankings. They range from the Condorcet solution to the one which is the most egalitarian with respect to the voters. We show that highly egalitarian rankings have the important property to be more stable with respect to fluctuations and that classical consensus rankings (Copeland, Tideman, Schulze) often turn out to be non optimal. The new dimension we have introduced provides, when used together with that of Condorcet, a clear classification of all the possible rankings. By increasing awareness in selecting a consensus ranking our method may lead to social choices which are more egalitarian compared to those achieved by presently available voting systems.

physics.soc-ph↗

arXiv2015

Cross correlations of the American baby names

The quantitative description of cultural evolution is a challenging task. The most difficult part of the problem is probably to find the appropriate measurable quantities that can make more quantitative such evasive concepts as, for example, dynamics of cultural movements, behavior patterns and traditions of the people. A strategy to tackle this issue is to observe particular features of human activities, i.e. cultural traits, such as names given to newborns. We study the names of babies born in the United States of America from 1910 to 2012. Our analysis shows that groups of different correlated states naturally emerge in different epochs, and we are able to follow and decrypt their evolution. While these groups of states are stable across many decades, a sudden reorganization occurs in the last part of the twentieth century. We think that this kind of quantitative analysis can be possibly extended to other cultural traits: although databases covering more than one century (as the one we used) are rare, the cultural evolution on shorter time scales can be studied thanks to the fact that many human activities are usually recorded in the present digital era.

physics.soc-ph↗

arXiv2015

Inferring Synaptic Structure in presence of Neural Interaction Time Scales

Biological networks display a variety of activity patterns reflecting a web of interactions that is complex both in space and time. Yet inference methods have mainly focused on reconstructing, from the network's activity, the spatial structure, by assuming equilibrium conditions or, more recently, a probabilistic dynamics with a single arbitrary time-step. Here we show that, under this latter assumption, the inference procedure fails to reconstruct the synaptic matrix of a network of integrate-and-fire neurons when the chosen time scale of interaction does not closely match the synaptic delay or when no single time scale for the interaction can be identified; such failure, moreover, exposes a distinctive bias of the inference method that can lead to infer as inhibitory the excitatory synapses with interaction time scales longer than the model's time-step. We therefore introduce a new two-step method, that first infers through cross-correlation profiles the delay-structure of the network and then reconstructs the synaptic matrix, and successfully test it on networks with different topologies and in different activity regimes. Although step one is able to accurately recover the delay-structure of the network, thus getting rid of any \textit{a priori} guess about the time scales of the interaction, the inference method introduces nonetheless an arbitrary time scale, the time-bin $dt$ used to binarize the spike trains. We therefore analytically and numerically study how the choice of $dt$ affects the inference in our network model, finding that the relationship between the inferred couplings and the real synaptic efficacies, albeit being quadratic in both cases, depends critically on $dt$ for the excitatory synapses only, whilst being basically independent of it for the inhibitory ones.

q-bio.NC↗

arXiv2015

Diluted Mean-Field Spin-Glass Models at Criticality

We present a method derived by cavity arguments to compute the spin-glass and higher-order susceptibilities in diluted mean-field spin-glass models. The divergence of the spin-glass susceptibility is associated to the existence of a non-zero solution of a homogeneous linear integral equation. Higher order susceptibilities, relevant for critical dynamics through the parameter exponent $λ$, can be expressed at criticality as integrals involving the critical eigenvector. The numerical evaluation of the corresponding analytic expressions is discussed. The method is illustrated in the context of the de Almeida-Thouless line for a spin-glass on a Bethe lattice but can be generalized straightforwardly to more complex situations.

cond-mat.dis-nn↗

arXiv2014

Boolean constraint satisfaction problems for reaction networks

We define and study a class of (random) Boolean constraint satisfaction problems representing minimal feasibility constraints for networks of chemical reactions. The constraints we consider encode, respectively, for hard mass-balance conditions (where the consumption and production fluxes of each chemical species are matched) and for soft mass-balance conditions (where a net production of compounds is in principle allowed). We solve these constraint satisfaction problems under the Bethe approximation and derive the corresponding Belief Propagation equations, that involve 8 different messages. The statistical properties of ensembles of random problems are studied via the population dynamics methods. By varying a chemical potential attached to the activity of reactions, we find first order transitions and strong hysteresis, suggesting a non-trivial structure in the space of feasible solutions.

q-bio.MN↗

arXiv2014

Finite size corrections to disordered Ising models on Random Regular Graphs

We derive the analytical expression for the first finite size correction to the average free energy of disordered Ising models on random regular graphs. The formula can be physically interpreted as a weighted sum over all non self-intersecting loops in the graph, the weight being the free-energy shift due to the addition of the loop to an infinite tree.

cond-mat.dis-nn↗

arXiv2014

The crossover region between long-range and short-range interactions for the critical exponents

It is well know that systems with an interaction decaying as a power of the distance may have critical exponents that are different from those of short-range systems. The boundary between long-range and short-range is known, however the behavior in the crossover region is not well understood. In this paper we propose a general form for the crossover function and we compute it in a particular limit. We compare our predictions with the results of numerical simulations for two-dimensional long-range percolation.

cond-mat.stat-mech↗

arXiv2014

Large Deviations of Correlation Functions in Random Magnets

We present a large deviations theory of the spin-spin correlation functions in the Random Field Ising Model on the Bethe lattice, both at finite and zero temperature. Rare events of atypically correlated variables are particularly important at the critical point: the phase transition is driven by few pairs of strongly correlated spins, while the majority remains basically uncorrelated. At the zero temperature critical point the number of spin pairs correlated over a distance L is shown to be no longer exponential, but only linear in the spins separation.

cond-mat.dis-nn↗

arXiv2014

Anomalous finite size corrections in random field models

The presence of a random magnetic field in ferromagnetic systems leads, in the broken phase, to an anomalous $O(\sqrt{1/N})$ convergence of some thermodynamic quantities to their asymptotic limits. Here we show a general method, based on the replica trick, to compute analytically the $O(\sqrt{1/N})$ finite size correction to the average free energy. We apply this method to two mean field Ising models, fully connected and random regular graphs, and compare the results to exact numerical algorithms. We argue that this behaviour is present in finite dimensional models as well.

cond-mat.dis-nn↗