arXiv subjects

Andrea Montanari

Publications and source records attributed to Andrea Montanari.

At least 217 records · Page 12Linked to original sources

arXiv2005

Tight bounds for LDPC and LDGM codes under MAP decoding

A new method for analyzing low density parity check (LDPC) codes and low density generator matrix (LDGM) codes under bit maximum a posteriori probability (MAP) decoding is introduced. The method is based on a rigorous approach to spin glasses developed by Francesco Guerra. It allows to construct lower bounds on the entropy of the transmitted message conditional to the received one. Based on heuristic statistical mechanics calculations, we conjecture such bounds to be tight. The result holds for standard irregular ensembles when used over binary input output symmetric channels. The method is first developed for Tanner graph ensembles with Poisson left degree distribution. It is then generalized to `multi-Poisson' graphs, and, by a completion procedure, to arbitrary degree distribution.

cs.IT↗

arXiv2005

From Large Scale Rearrangements to Mode Coupling Phenomenology

We consider the equilibrium dynamics of Ising spin models with multi-spin interactions on sparse random graphs (Bethe lattices). Such models undergo a mean field glass transition upon increasing the graph connectivity or lowering the temperature. Focusing on the low temperature limit, we identify the large scale rearrangements responsible for the dynamical slowing-down near the transition. We are able to characterize exactly the dynamics near criticality by analyzing the statistical properties of such rearrangements. Our approach can be generalized to a large variety of glassy models on sparse random graphs, ranging from satisfiability to kinetically constrained models.

cond-mat.stat-mech↗

arXiv2004

Life Above Threshold: From List Decoding to Area Theorem and MSE

We consider communication over memoryless channels using low-density parity-check code ensembles above the iterative (belief propagation) threshold. What is the computational complexity of decoding (i.e., of reconstructing all the typical input codewords for a given channel output) in this regime? We define an algorithm accomplishing this task and analyze its typical performance. The behavior of the new algorithm can be expressed in purely information-theoretical terms. Its analysis provides an alternative proof of the area theorem for the binary erasure channel. Finally, we explain how the area theorem is generalized to arbitrary memoryless channels. We note that the recently discovered relation between mutual information and minimal square error is an instance of the area theorem in the setting of Gaussian channels.

cs.IT↗

arXiv2004

Finite-Length Scaling and Finite-Length Shift for Low-Density Parity-Check Codes

Consider communication over the binary erasure channel BEC using random low-density parity-check codes with finite-blocklength n from `standard' ensembles. We show that large error events is conveniently described within a scaling theory, and explain how to estimate heuristically their effect. Among other quantities, we consider the finite length threshold e(n), defined by requiring a block error probability P_B = 1/2. For ensembles with minimum variable degree larger than two, the following expression is argued to hold e(n) = e -e_1 n^{-2/3} +Θ(n^{-1}) with a calculable shift} parameter e_1>0.

cs.IT↗

arXiv2004

On the cooling-schedule dependence of the dynamics of mean-field glasses

The low temperature phase of discontinuous mean-field spin glasses is characterized by the appearance of an exponential number of metastable states. Which ones among these states dominate the out-of-equilibrium dynamics of these systems? In order to answer this question, we compare high-precision numerical simulations of a diluted $p$-spin model with a cavity computation of the threshold energy. Our main conclusion is that the aging dynamics is dominated by different layers of metastable states depending on the cooling schedule. In order to perform our analysis, we present a method for computing the marginality condition of diluted spin glasses at non-zero temperature.

cond-mat.dis-nn↗

arXiv2004

Finite-Length Scaling for Iteratively Decoded LDPC Ensembles

In this paper we investigate the behavior of iteratively decoded low-density parity-check codes over the binary erasure channel in the so-called ``waterfall region." We show that the performance curves in this region follow a very basic scaling law. We conjecture that essentially the same scaling behavior applies in a much more general setting and we provide some empirical evidence to support this conjecture. The scaling law, together with the error floor expressions developed previously, can be used for fast finite-length optimization.

cs.IT↗

arXiv2003

Instability of one-step replica-symmetry-broken phase in satisfiability problems

We reconsider the one-step replica-symmetry-breaking (1RSB) solutions of two random combinatorial problems: k-XORSAT and k-SAT. We present a general method for establishing the stability of these solutions with respect to further steps of replica-symmetry breaking. Our approach extends the ideas of [A.Montanari and F. Ricci-Tersenghi, Eur.Phys.J. B 33, 339 (2003)] to more general combinatorial problems. It turns out that 1RSB is always unstable at sufficiently small clauses density alpha or high energy. In particular, the recent 1RSB solution to 3-SAT is unstable at zero energy for alpha< alpha_m, with alpha_m\approx 4.153. On the other hand, the SAT-UNSAT phase transition seems to be correctly described within 1RSB.

cond-mat.dis-nn↗

arXiv2003

Aging dynamics of heterogeneous spin models

We investigate numerically the dynamics of three different spin models in the aging regime. Each of these models is meant to be representative of a distinct class of aging behavior: coarsening systems, discontinuous spin glasses, and continuous spin glasses. In order to study dynamic heterogeneities induced by quenched disorder, we consider single-spin observables for a given disorder realization. In some simple cases we are able to provide analytical predictions for single-spin response and correlation functions. The results strongly depend upon the model considered. It turns out that, by comparing the slow evolution of a few different degrees of freedom, one can distinguish between different dynamic classes. As a conclusion we present the general properties which can be induced from our results, and discuss their relation with thermometric arguments.

cond-mat.dis-nn↗

arXiv2003

On the stochastic dynamics of disordered spin models

In this article we discuss several aspects of the stochastic dynamics of spin models. The paper has two independent parts. Firstly, we explore a few properties of the multi-point correlations and responses of generic systems evolving in equilibrium with a thermal bath. We propose a fluctuation principle that allows us to derive fluctuation-dissipation relations for many-time correlations and linear responses. We also speculate on how these features will be modified in systems evolving slowly out of equilibrium, as finite-dimensional or dilute spin-glasses. Secondly, we present a formalism that allows one to derive a series of approximated equations that determine the dynamics of disordered spin models on random (hyper) graphs.

cond-mat.stat-mech↗

arXiv2003

Approximate analysis of search algorithms with "physical" methods

An overview of some methods of statistical physics applied to the analysis of algorithms for optimization problems (satisfiability of Boolean constraints, vertex cover of graphs, decoding, ...) with distributions of random inputs is proposed. Two types of algorithms are analyzed: complete procedures with backtracking (Davis-Putnam-Loveland-Logeman algorithm) and incomplete, local search procedures (gradient descent, random walksat, ...). The study of complete algorithms makes use of physical concepts such as phase transitions, dynamical renormalization flow, growth processes, ... As for local search procedures, the connection between computational complexity and the structure of the cost function landscape is questioned, with emphasis on the notion of metastability.

cs.CC↗

arXiv2003

On the nature of the low-temperature phase in discontinuous mean-field spin glasses

The low-temperature phase of discontinuous mean-field spin glasses is generally described by a one-step replica symmetry breaking (1RSB) Ansatz. The Gardner transition, i.e. a very-low-temperature phase transition to a full replica symmetry breaking (FRSB) phase, is often regarded as an inessential, and somehow exotic phenomenon. In this paper we show that the metastable states which are relevant for the out-of-equilibrium dynamics of such systems are always in a FRSB phase. The only exceptions are (to the best of our knowledge) the p-spin spherical model and the random energy model (REM). We also discuss the consequences of our results for aging dynamics and for local search algorithms in hard combinatorial problems.

cond-mat.dis-nn↗

arXiv2002

A microscopic description of the aging dynamics: fluctuation-dissipation relations, effective temperature and heterogeneities

We consider the dynamics of a diluted mean-field spin glass model in the aging regime. The model presents a particularly rich heterogeneous behavior. In order to catch this behavior, we perform a **spin-by-spin analysis** for a **given disorder realization**. The results compare well with the outcome of a static calculation which uses the ``survey propagation'' algorithm of Mezard, Parisi, and Zecchina [Sciencexpress 10.1126/science.1073287 (2002)]. We thus confirm the connection between statics and dynamics at the level of single degrees of freedom. Moreover, working with single-site quantities, we can introduce a new response-vs-correlation plot, which clearly shows how heterogeneous degrees of freedom undergo coherent structural rearrangements. Finally we discuss the general scenario which emerges from our work and (possibly) applies to more realistic glassy models. Interestingly enough, some features of this scenario can be understood recurring to thermometric considerations.

cond-mat.dis-nn↗

arXiv2002

The Dynamic Phase Transition for Decoding Algorithms

The state-of-the-art error correcting codes are based on large random constructions (random graphs, random permutations, ...) and are decoded by linear-time iterative algorithms. Because of these features, they are remarkable examples of diluted mean-field spin glasses, both from the static and from the dynamic points of view. We analyze the behavior of decoding algorithms using the mapping onto statistical-physics models. This allows to understand the intrinsic (i.e. algorithm independent) features of this behavior.

cond-mat.dis-nn↗

arXiv2001

Boosting search by rare events

Randomized search algorithms for hard combinatorial problems exhibit a large variability of performances. We study the different types of rare events which occur in such out-of-equilibrium stochastic processes and we show how they cooperate in determining the final distribution of running times. As a byproduct of our analysis we show how search algorithms are optimized by random restarts.

cond-mat.stat-mech↗

arXiv2001

Discrete non-Abelian groups and asymptotically free models

We consider a two-dimensional $σ$-model with discrete icosahedral/dodecahedral symmetry. Using the perturbative renormalization group, we argue that this model has a different continuum limit with respect to the O(3) $σ$ model. Such an argument is confirmed by a high-precision numerical simulation.

hep-lat↗

arXiv2001

Spin models on Platonic solids and asymptotic freedom

We consider a two-dimensional sigma-model with discrete icosahedral/dodecahedral symmetry. We present high-precision finite-size numerical results that show that the continuum limit of this model is different from the continuum limit of the rotationally invariant O(3) sigma-model.

hep-lat↗

arXiv2001

Asymptotically free models and discrete non-Abelian groups

We study the two-dimensional renormalization-group flow induced by perturbations that reduce the global symmetry of the O(3) sigma-model to the discrete symmetries of Platonic solids. We estimate the value of the correlation length at which differences in the behaviour of the various models should be expected. For the icosahedron model, we find xi > 200. We provide an explanation for the recent numerical results of Patrascioiu and Seiler and of Hasenfratz and Niedermayer.

hep-lat↗

arXiv2001

Non-Perturbative Renormalization in Lattice Field Theory

Non-perturbative renormalization of lattice composite operators plays a crucial role in many applications of lattice field theory. We sketch the general problems involved in this task and the methods which are currently used to cope with them. We present a detailed investigation of a new approach based on the operator product expansion. We test the new method on the two-dimensional O(3) sigma-model and discuss its advantages and limitations. Ph.D. thesis at Scuola Normale Superiore, Pisa, Italy.

hep-lat↗