Gaussian Belief with dynamic data and in dynamic network

In this paper we analyse Belief Propagation over a Gaussian model in a dynamic environment. Recently, this has been proposed as a method to average local measurement values by a distributed protocol ("Consensus Propagation", Moallemi & Van Roy, 2006), where the average is available for read-out at every single node. In the case that the underlying network is constant but the values to be averaged fluctuate ("dynamic data"), convergence and accuracy are determined by the spectral properties of an associated Ruelle-Perron-Frobenius operator. For Gaussian models on Erdos-Renyi graphs, numerical computation points to a spectral gap remaining in the large-size limit, implying exceptionally good scalability. In a model where the underlying network also fluctuates ("dynamic network"), averaging is more effective than in the dynamic data case. Altogether, this implies very good performance of these methods in very large systems, and opens a new field of statistical physics of large (and dynamic) information systems.Comment: 5 pages, 7 figure

Hierarchical Models for Independence Structures of Networks

We introduce a new family of network models, called hierarchical network models, that allow us to represent in an explicit manner the stochastic dependence among the dyads (random ties) of the network. In particular, each member of this family can be associated with a graphical model defining conditional independence clauses among the dyads of the network, called the dependency graph. Every network model with dyadic independence assumption can be generalized to construct members of this new family. Using this new framework, we generalize the Erd\"os-R\'enyi and beta-models to create hierarchical Erd\"os-R\'enyi and beta-models. We describe various methods for parameter estimation as well as simulation studies for models with sparse dependency graphs.Comment: 19 pages, 7 figure

Atmosphere, Interior, and Evolution of the Metal-Rich Transiting Planet HD 149026b

We investigate the atmosphere and interior of the new transiting planet HD 149026b, which appears to be very rich in heavy elements. We first compute model atmospheres at metallicities ranging from solar to ten times solar, and show how for cases with high metallicity or inefficient redistribution of energy from the day side, the planet may develop a hot stratosphere due to absorption of stellar flux by TiO and VO. The spectra predicted by these models are very different than cooler atmosphere models without stratospheres. The spectral effects are potentially detectable with the Spitzer Space Telescope. In addition the models with hot stratospheres lead to a large limb brightening, rather than darkening. We compare the atmosphere of HD 149026b to other well-known transiting planets, including the recently discovered HD 189733b, which we show have planet-to-star flux ratios twice that of HD 209458 and TrES-1. The methane abundance in the atmosphere of HD 189733b is a sensitive indicator of atmospheric temperature and metallicity and can be constrained with Spitzer IRAC observations. We then turn to interior studies of HD 149026b and use a grid of self-consistent model atmospheres and high-pressure equations of state for all components to compute thermal evolution models of the planet. We estimate that the mass of heavy elements within the planet is in the range of 60 to 93 M_earth. Finally, we discuss trends in the radii of transiting planets with metallicity in light of this new member of the class.Comment: Accepted to the Astrophysical Journal. 18 pages, including 10 figures. New section on the atmosphere of planet HD 189733b. Enhanced discussion of atmospheric Ti chemistry and core mass for HD 149026

Colouring random graphs and maximising local diversity

We study a variation of the graph colouring problem on random graphs of finite average connectivity. Given the number of colours, we aim to maximise the number of different colours at neighbouring vertices (i.e. one edge distance) of any vertex. Two efficient algorithms, belief propagation and Walksat are adapted to carry out this task. We present experimental results based on two types of random graphs for different system sizes and identify the critical value of the connectivity for the algorithms to find a perfect solution. The problem and the suggested algorithms have practical relevance since various applications, such as distributed storage, can be mapped onto this problem.Comment: 10 pages, 10 figure

Dispersion control for matter waves and gap solitons in optical superlattices

We present a numerical study of dispersion manipulation and formation of matter-wave gap solitons in a Bose-Einstein condensate trapped in an optical superlattice. We demonstrate a method for controlled generation of matter-wave gap solitons in a stationary lattice by using an interference pattern of two condensate wavepackets, which mimics the structure of the gap soliton near the edge of a spectral band. The efficiency of this method is compared with that of gap soliton generation in a moving lattice recently demonstrated experimentally by Eiermann et al. [Phys. Rev. Lett. 92, 230401 (2004)]. We show that, by changing the relative depths of the superlattice wells, one can fine-tune the effective dispersion of the matter waves at the edges of the mini-gaps of the superlattice Bloch-wave spectrum and therefore effectively control both the peak density and the spatial width of the emerging gap solitons.Comment: 8 pages, 9 figures; modified references in Section 2; minor content changes in Sections 1 and 2 and Fig. 9 captio

The Phase Diagram of 1-in-3 Satisfiability Problem

We study the typical case properties of the 1-in-3 satisfiability problem, the boolean satisfaction problem where a clause is satisfied by exactly one literal, in an enlarged random ensemble parametrized by average connectivity and probability of negation of a variable in a clause. Random 1-in-3 Satisfiability and Exact 3-Cover are special cases of this ensemble. We interpolate between these cases from a region where satisfiability can be typically decided for all connectivities in polynomial time to a region where deciding satisfiability is hard, in some interval of connectivities. We derive several rigorous results in the first region, and develop the one-step--replica-symmetry-breaking cavity analysis in the second one. We discuss the prediction for the transition between the almost surely satisfiable and the almost surely unsatisfiable phase, and other structural properties of the phase diagram, in light of cavity method results.Comment: 30 pages, 12 figure

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

Inference by replication in densely connected systems

An efficient Bayesian inference method for problems that can be mapped onto dense graphs is presented. The approach is based on message passing where messages are averaged over a large number of replicated variable systems exposed to the same evidential nodes. An assumption about the symmetry of the solutions is required for carrying out the averages; here we extend the previous derivation based on a replica symmetric (RS) like structure to include a more complex one-step replica symmetry breaking (1RSB)-like ansatz. To demonstrate the potential of the approach it is employed for studying critical properties of the Ising linear perceptron and for multiuser detection in Code Division Multiple Access (CDMA) under different noise models. Results obtained under the RS assumption in the non-critical regime give rise to a highly efficient signal detection algorithm in the context of CDMA; while in the critical regime one observes a first order transition line that ends in a continuous phase transition point. Finite size effects are also observed. While the 1RSB ansatz is not required for the original problems, it was applied to the CDMA signal detection problem with a more complex noise model that exhibits RSB behaviour, resulting in an improvement in performance.Comment: 47 pages, 7 figure

Binary Models for Marginal Independence

Log-linear models are a classical tool for the analysis of contingency tables. In particular, the subclass of graphical log-linear models provides a general framework for modelling conditional independences. However, with the exception of special structures, marginal independence hypotheses cannot be accommodated by these traditional models. Focusing on binary variables, we present a model class that provides a framework for modelling marginal independences in contingency tables. The approach taken is graphical and draws on analogies to multivariate Gaussian models for marginal independence. For the graphical model representation we use bi-directed graphs, which are in the tradition of path diagrams. We show how the models can be parameterized in a simple fashion, and how maximum likelihood estimation can be performed using a version of the Iterated Conditional Fitting algorithm. Finally we consider combining these models with symmetry restrictions

Contextual Object Detection with a Few Relevant Neighbors

A natural way to improve the detection of objects is to consider the contextual constraints imposed by the detection of additional objects in a given scene. In this work, we exploit the spatial relations between objects in order to improve detection capacity, as well as analyze various properties of the contextual object detection problem. To precisely calculate context-based probabilities of objects, we developed a model that examines the interactions between objects in an exact probabilistic setting, in contrast to previous methods that typically utilize approximations based on pairwise interactions. Such a scheme is facilitated by the realistic assumption that the existence of an object in any given location is influenced by only few informative locations in space. Based on this assumption, we suggest a method for identifying these relevant locations and integrating them into a mostly exact calculation of probability based on their raw detector responses. This scheme is shown to improve detection results and provides unique insights about the process of contextual inference for object detection. We show that it is generally difficult to learn that a particular object reduces the probability of another, and that in cases when the context and detector strongly disagree this learning becomes virtually impossible for the purposes of improving the results of an object detector. Finally, we demonstrate improved detection results through use of our approach as applied to the PASCAL VOC and COCO datasets