Error correcting code using tree-like multilayer perceptron

An error correcting code using a tree-like multilayer perceptron is proposed. An original message \mbi{s}^0 is encoded into a codeword \boldmath{y}_0 using a tree-like committee machine (committee tree) or a tree-like parity machine (parity tree). Based on these architectures, several schemes featuring monotonic or non-monotonic units are introduced. The codeword \mbi{y}_0 is then transmitted via a Binary Asymmetric Channel (BAC) where it is corrupted by noise. The analytical performance of these schemes is investigated using the replica method of statistical mechanics. Under some specific conditions, some of the proposed schemes are shown to saturate the Shannon bound at the infinite codeword length limit. The influence of the monotonicity of the units on the performance is also discussed.Comment: 23 pages, 3 figures, Content has been extended and revise

Towards gravitationally assisted negative refraction of light by vacuum

Propagation of electromagnetic plane waves in some directions in gravitationally affected vacuum over limited ranges of spacetime can be such that the phase velocity vector casts a negative projection on the time-averaged Poynting vector. This conclusion suggests, inter alia, gravitationally assisted negative refraction by vacuum.Comment: 6 page

Measuring the effective complexity of cosmological models

We introduce a statistical measure of the effective model complexity, called the Bayesian complexity. We demonstrate that the Bayesian complexity can be used to assess how many effective parameters a set of data can support and that it is a useful complement to the model likelihood (the evidence) in model selection questions. We apply this approach to recent measurements of cosmic microwave background anisotropies combined with the Hubble Space Telescope measurement of the Hubble parameter. Using mildly non-informative priors, we show how the 3-year WMAP data improves on the first-year data by being able to measure both the spectral index and the reionization epoch at the same time. We also find that a non-zero curvature is strongly disfavored. We conclude that although current data could constrain at least seven effective parameters, only six of them are required in a scheme based on the Lambda-CDM concordance cosmology.Comment: 9 pages, 4 figures, revised version accepted for publication in PRD, updated with WMAP3 result

Heteroclinic intersections between Invariant Circles of Volume-Preserving Maps

We develop a Melnikov method for volume-preserving maps with codimension one invariant manifolds. The Melnikov function is shown to be related to the flux of the perturbation through the unperturbed invariant surface. As an example, we compute the Melnikov function for a perturbation of a three-dimensional map that has a heteroclinic connection between a pair of invariant circles. The intersection curves of the manifolds are shown to undergo bifurcations in homologyComment: LaTex with 10 eps figure

Bayesian stochastic blockmodeling

This chapter provides a self-contained introduction to the use of Bayesian inference to extract large-scale modular structures from network data, based on the stochastic blockmodel (SBM), as well as its degree-corrected and overlapping generalizations. We focus on nonparametric formulations that allow their inference in a manner that prevents overfitting, and enables model selection. We discuss aspects of the choice of priors, in particular how to avoid underfitting via increased Bayesian hierarchies, and we contrast the task of sampling network partitions from the posterior distribution with finding the single point estimate that maximizes it, while describing efficient algorithms to perform either one. We also show how inferring the SBM can be used to predict missing and spurious links, and shed light on the fundamental limitations of the detectability of modular structures in networks.Comment: 44 pages, 16 figures. Code is freely available as part of graph-tool at https://graph-tool.skewed.de . See also the HOWTO at https://graph-tool.skewed.de/static/doc/demos/inference/inference.htm

Quantum random walks with decoherent coins

The quantum random walk has been much studied recently, largely due to its highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum walk on the line: the presence of decoherence in the quantum ``coin'' which drives the walk. We find exact analytical expressions for the time dependence of the first two moments of position, and show that in the long-time limit the variance grows linearly with time, unlike the unitary walk. We compare this to the results of direct numerical simulation, and see how the form of the position distribution changes from the unitary to the usual classical result as we increase the strength of the decoherence.Comment: Minor revisions, especially in introduction. Published versio

On Cavity Approximations for Graphical Models

We reformulate the Cavity Approximation (CA), a class of algorithms recently introduced for improving the Bethe approximation estimates of marginals in graphical models. In our new formulation, which allows for the treatment of multivalued variables, a further generalization to factor graphs with arbitrary order of interaction factors is explicitly carried out, and a message passing algorithm that implements the first order correction to the Bethe approximation is described. Furthermore we investigate an implementation of the CA for pairwise interactions. In all cases considered we could confirm that CA[k] with increasing $k$ provides a sequence of approximations of markedly increasing precision. Furthermore in some cases we could also confirm the general expectation that the approximation of order $k$, whose computational complexity is $O(N^{k+1})$ has an error that scales as $1/N^{k+1}$ with the size of the system. We discuss the relation between this approach and some recent developments in the field.Comment: Extension to factor graphs and comments on related work adde

Statistical mechanical aspects of joint source-channel coding

An MN-Gallager Code over Galois fields, $q$, based on the Dynamical Block Posterior probabilities (DBP) for messages with a given set of autocorrelations is presented with the following main results: (a) for a binary symmetric channel the threshold, $f_c$, is extrapolated for infinite messages using the scaling relation for the median convergence time, $t_{med} \propto 1/(f_c-f)$; (b) a degradation in the threshold is observed as the correlations are enhanced; (c) for a given set of autocorrelations the performance is enhanced as $q$ is increased; (d) the efficiency of the DBP joint source-channel coding is slightly better than the standard gzip compression method; (e) for a given entropy, the performance of the DBP algorithm is a function of the decay of the correlation function over large distances.Comment: 6 page

The Statistical Physics of Regular Low-Density Parity-Check Error-Correcting Codes

A variation of Gallager error-correcting codes is investigated using statistical mechanics. In codes of this type, a given message is encoded into a codeword which comprises Boolean sums of message bits selected by two randomly constructed sparse matrices. The similarity of these codes to Ising spin systems with random interaction makes it possible to assess their typical performance by analytical methods developed in the study of disordered systems. The typical case solutions obtained via the replica method are consistent with those obtained in simulations using belief propagation (BP) decoding. We discuss the practical implications of the results obtained and suggest a computationally efficient construction for one of the more practical configurations.Comment: 35 pages, 4 figure

Ground states and formal duality relations in the Gaussian core model

We study dimensional trends in ground states for soft-matter systems. Specifically, using a high-dimensional version of Parrinello-Rahman dynamics, we investigate the behavior of the Gaussian core model in up to eight dimensions. The results include unexpected geometric structures, with surprising anisotropy as well as formal duality relations. These duality relations suggest that the Gaussian core model possesses unexplored symmetries, and they have implications for a broad range of soft-core potentials.Comment: 7 pages, 1 figure, appeared in Physical Review E (http://pre.aps.org