Convergence Rate Analysis of Distributed Gossip (Linear Parameter) Estimation: Fundamental Limits and Tradeoffs

The paper considers gossip distributed estimation of a (static) distributed random field (a.k.a., large scale unknown parameter vector) observed by sparsely interconnected sensors, each of which only observes a small fraction of the field. We consider linear distributed estimators whose structure combines the information \emph{flow} among sensors (the \emph{consensus} term resulting from the local gossiping exchange among sensors when they are able to communicate) and the information \emph{gathering} measured by the sensors (the \emph{sensing} or \emph{innovations} term.) This leads to mixed time scale algorithms--one time scale associated with the consensus and the other with the innovations. The paper establishes a distributed observability condition (global observability plus mean connectedness) under which the distributed estimates are consistent and asymptotically normal. We introduce the distributed notion equivalent to the (centralized) Fisher information rate, which is a bound on the mean square error reduction rate of any distributed estimator; we show that under the appropriate modeling and structural network communication conditions (gossip protocol) the distributed gossip estimator attains this distributed Fisher information rate, asymptotically achieving the performance of the optimal centralized estimator. Finally, we study the behavior of the distributed gossip estimator when the measurements fade (noise variance grows) with time; in particular, we consider the maximum rate at which the noise variance can grow and still the distributed estimator being consistent, by showing that, as long as the centralized estimator is consistent, the distributed estimator remains consistent.Comment: Submitted for publication, 30 page

Gossip and Distributed Kalman Filtering: Weak Consensus under Weak Detectability

The paper presents the gossip interactive Kalman filter (GIKF) for distributed Kalman filtering for networked systems and sensor networks, where inter-sensor communication and observations occur at the same time-scale. The communication among sensors is random; each sensor occasionally exchanges its filtering state information with a neighbor depending on the availability of the appropriate network link. We show that under a weak distributed detectability condition: 1. the GIKF error process remains stochastically bounded, irrespective of the instability properties of the random process dynamics; and 2. the network achieves \emph{weak consensus}, i.e., the conditional estimation error covariance at a (uniformly) randomly selected sensor converges in distribution to a unique invariant measure on the space of positive semi-definite matrices (independent of the initial state.) To prove these results, we interpret the filtered states (estimates and error covariances) at each node in the GIKF as stochastic particles with local interactions. We analyze the asymptotic properties of the error process by studying as a random dynamical system the associated switched (random) Riccati equation, the switching being dictated by a non-stationary Markov chain on the network graph.Comment: Submitted to the IEEE Transactions, 30 pages

Electric field dynamics and ion acceleration in the self-channeling of a superintense laser pulse

The dynamics of electric field generation and radial acceleration of ions by a laser pulse of relativistic intensity propagating in an underdense plasma has been investigated using an one-dimensional electrostatic, ponderomotive model developed to interpret experimental measurements of electric fields [S. Kar et al, New J. Phys. *9*, 402 (2007)]. Ions are spatially focused at the edge of the charge-displacement channel, leading to hydrodynamical breaking, which in turns causes the heating of electrons and an "echo" effect in the electric field. The onset of complete electron depletion in the central region of the channel leads to a smooth transition to a "Coulomb explosion" regime and a saturation of ion acceleration.Comment: 9 pages, 7 figures, final revised version, to appear on Plasma Phys. Contr. Fus., special issue on "Laser and Plasma Accelerators", scheduled for February, 200

Sensor Networks with Random Links: Topology Design for Distributed Consensus

In a sensor network, in practice, the communication among sensors is subject to:(1) errors or failures at random times; (3) costs; and(2) constraints since sensors and networks operate under scarce resources, such as power, data rate, or communication. The signal-to-noise ratio (SNR) is usually a main factor in determining the probability of error (or of communication failure) in a link. These probabilities are then a proxy for the SNR under which the links operate. The paper studies the problem of designing the topology, i.e., assigning the probabilities of reliable communication among sensors (or of link failures) to maximize the rate of convergence of average consensus, when the link communication costs are taken into account, and there is an overall communication budget constraint. To consider this problem, we address a number of preliminary issues: (1) model the network as a random topology; (2) establish necessary and sufficient conditions for mean square sense (mss) and almost sure (a.s.) convergence of average consensus when network links fail; and, in particular, (3) show that a necessary and sufficient condition for both mss and a.s. convergence is for the algebraic connectivity of the mean graph describing the network topology to be strictly positive. With these results, we formulate topology design, subject to random link failures and to a communication cost constraint, as a constrained convex optimization problem to which we apply semidefinite programming techniques. We show by an extensive numerical study that the optimal design improves significantly the convergence speed of the consensus algorithm and can achieve the asymptotic performance of a non-random network at a fraction of the communication cost.Comment: Submitted to IEEE Transaction

Distributed Convergence Verification for Gaussian Belief Propagation

Gaussian belief propagation (BP) is a computationally efficient method to approximate the marginal distribution and has been widely used for inference with high dimensional data as well as distributed estimation in large-scale networks. However, the convergence of Gaussian BP is still an open issue. Though sufficient convergence conditions have been studied in the literature, verifying these conditions requires gathering all the information over the whole network, which defeats the main advantage of distributed computing by using Gaussian BP. In this paper, we propose a novel sufficient convergence condition for Gaussian BP that applies to both the pairwise linear Gaussian model and to Gaussian Markov random fields. We show analytically that this sufficient convergence condition can be easily verified in a distributed way that satisfies the network topology constraint.Comment: accepted by Asilomar Conference on Signals, Systems, and Computers, 2017, Asilomar, Pacific Grove, CA. arXiv admin note: text overlap with arXiv:1706.0407