Networked Slepian-Wolf: theory, algorithms, and scaling laws

Consider a set of correlated sources located at the nodes of a network, and a set of sinks that are the destinations for some of the sources. The minimization of cost functions which are the product of a function of the rate and a function of the path weight is considered, for both the data-gathering scenario, which is relevant in sensor networks, and general traffic matrices, relevant for general networks. The minimization is achieved by jointly optimizing a) the transmission structure, which is shown to consist in general of a superposition of trees, and b) the rate allocation across the source nodes, which is done by Slepian-Wolf coding. The overall minimization can be achieved in two concatenated steps. First, the optimal transmission structure is found, which in general amounts to finding a Steiner tree, and second, the optimal rate allocation is obtained by solving an optimization problem with cost weights determined by the given optimal transmission structure, and with linear constraints given by the Slepian-Wolf rate region. For the case of data gathering, the optimal transmission structure is fully characterized and a closed-form solution for the optimal rate allocation is provided. For the general case of an arbitrary traffic matrix, the problem of finding the optimal transmission structure is NP-complete. For large networks, in some simplified scenarios, the total costs associated with Slepian-Wolf coding and explicit communication (conditional encoding based on explicitly communicated side information) are compared. Finally, the design of decentralized algorithms for the optimal rate allocation is analyzed

A constructive mean field analysis of multi population neural networks with random synaptic weights and stochastic inputs

We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials. They are coupled by synaptic connections acting on their resulting activity, a nonlinear function of their membrane potential. At the second (mesoscopic) scale, interacting populations of neurons are described individually by similar equations. The equations describing the dynamical and the stationary mean field behaviors are considered as functional equations on a set of stochastic processes. Using this new point of view allows us to prove that these equations are well-posed on any finite time interval and to provide a constructive method for effectively computing their unique solution. This method is proved to converge to the unique solution and we characterize its complexity and convergence rate. We also provide partial results for the stationary problem on infinite time intervals. These results shed some new light on such neural mass models as the one of Jansen and Rit \cite{jansen-rit:95}: their dynamics appears as a coarse approximation of the much richer dynamics that emerges from our analysis. Our numerical experiments confirm that the framework we propose and the numerical methods we derive from it provide a new and powerful tool for the exploration of neural behaviors at different scales.Comment: 55 pages, 4 figures, to appear in "Frontiers in Neuroscience

On transient dynamics, off-equilibrium behaviour and identification in blended multiple model structures

The use of multiple-model techniques has been reported in a variety of control and signal processing applications. However, several theoretical analyses have recently appeared which outline fundamental limitations of these techniques in certain domains of application. In particular, the identifiability and interpretability of local linear model parameters in transient operating regimes is shown to be limited. Some modifications to the basic paradigm are suggested which overcome a number of problems. As an alternative to parametric identification of blended multiple model structures, nonparametric Gaussian process priors are suggested as a means of providing local models, and the results compared to a multiple-model approach in a Monte Carlo simulation on some simulated vehicle dynamics data

Steady-state simulation of reflected Brownian motion and related stochastic networks

This paper develops the first class of algorithms that enable unbiased estimation of steady-state expectations for multidimensional reflected Brownian motion. In order to explain our ideas, we first consider the case of compound Poisson (possibly Markov modulated) input. In this case, we analyze the complexity of our procedure as the dimension of the network increases and show that, under certain assumptions, the algorithm has polynomial-expected termination time. Our methodology includes procedures that are of interest beyond steady-state simulation and reflected processes. For instance, we use wavelets to construct a piecewise linear function that can be guaranteed to be within $\varepsilon$ distance (deterministic) in the uniform norm to Brownian motion in any compact time interval.Comment: Published at http://dx.doi.org/10.1214/14-AAP1072 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic Exit Location Distributions in the Stochastic Exit Problem

Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point $S$. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength $\epsilon$, the system state will eventually leave the domain of attraction $\Omega$ of $S$. We analyse the case when, as $\epsilon\to0$, the exit location on the boundary $\partial\Omega$ is increasingly concentrated near a saddle point $H$ of the deterministic dynamics. We show that the asymptotic form of the exit location distribution on $\partial\Omega$ is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter $\mu$, equal to the ratio $|\lambda_s(H)|/\lambda_u(H)$ of the stable and unstable eigenvalues of the linearized deterministic flow at $H$. If $\mu<1$ then the exit location distribution is generically asymptotic as $\epsilon\to0$ to a Weibull distribution with shape parameter $2/\mu$, on the $O(\epsilon^{\mu/2})$ length scale near $H$. If $\mu>1$ it is generically asymptotic to a distribution on the $O(\epsilon^{1/2})$ length scale, whose moments we compute. The asymmetry of the asymptotic exit location distribution is attributable to the generic presence of a `classically forbidden' region: a wedge-shaped subset of $\Omega$ with $H$ as vertex, which is reached from $S$, in the $\epsilon\to0$ limit, only via `bent' (non-smooth) fluctuational paths that first pass through the vicinity of $H$. We deduce from the presence of this forbidden region that the classical Eyring formula for the small-$\epsilon$ exponential asymptotics of the mean first exit time is generically inapplicable.Comment: This is a 72-page Postscript file, about 600K in length. Hardcopy requests to [email protected] or [email protected]

Practical Bayesian Modeling and Inference for Massive Spatial Datasets On Modest Computing Environments

With continued advances in Geographic Information Systems and related computational technologies, statisticians are often required to analyze very large spatial datasets. This has generated substantial interest over the last decade, already too vast to be summarized here, in scalable methodologies for analyzing large spatial datasets. Scalable spatial process models have been found especially attractive due to their richness and flexibility and, particularly so in the Bayesian paradigm, due to their presence in hierarchical model settings. However, the vast majority of research articles present in this domain have been geared toward innovative theory or more complex model development. Very limited attention has been accorded to approaches for easily implementable scalable hierarchical models for the practicing scientist or spatial analyst. This article is submitted to the Practice section of the journal with the aim of developing massively scalable Bayesian approaches that can rapidly deliver Bayesian inference on spatial process that are practically indistinguishable from inference obtained using more expensive alternatives. A key emphasis is on implementation within very standard (modest) computing environments (e.g., a standard desktop or laptop) using easily available statistical software packages without requiring message-parsing interfaces or parallel programming paradigms. Key insights are offered regarding assumptions and approximations concerning practical efficiency.Comment: 20 pages, 4 figures, 2 table