Chebyshev Polynomial Approximation for Distributed Signal Processing

Unions of graph Fourier multipliers are an important class of linear operators for processing signals defined on graphs. We present a novel method to efficiently distribute the application of these operators to the high-dimensional signals collected by sensor networks. The proposed method features approximations of the graph Fourier multipliers by shifted Chebyshev polynomials, whose recurrence relations make them readily amenable to distributed computation. We demonstrate how the proposed method can be used in a distributed denoising task, and show that the communication requirements of the method scale gracefully with the size of the network.Comment: 8 pages, 5 figures, to appear in the Proceedings of the IEEE International Conference on Distributed Computing in Sensor Systems (DCOSS), June, 2011, Barcelona, Spai

Accelerated Gossip in Networks of Given Dimension using Jacobi Polynomial Iterations

Consider a network of agents connected by communication links, where each agent holds a real value. The gossip problem consists in estimating the average of the values diffused in the network in a distributed manner. We develop a method solving the gossip problem that depends only on the spectral dimension of the network, that is, in the communication network set-up, the dimension of the space in which the agents live. This contrasts with previous work that required the spectral gap of the network as a parameter, or suffered from slow mixing. Our method shows an important improvement over existing algorithms in the non-asymptotic regime, i.e., when the values are far from being fully mixed in the network. Our approach stems from a polynomial-based point of view on gossip algorithms, as well as an approximation of the spectral measure of the graphs with a Jacobi measure. We show the power of the approach with simulations on various graphs, and with performance guarantees on graphs of known spectral dimension, such as grids and random percolation bonds. An extension of this work to distributed Laplacian solvers is discussed. As a side result, we also use the polynomial-based point of view to show the convergence of the message passing algorithm for gossip of Moallemi \& Van Roy on regular graphs. The explicit computation of the rate of the convergence shows that message passing has a slow rate of convergence on graphs with small spectral gap

Which Digraphs with Ring Structure are Essentially Cyclic?

We say that a digraph is essentially cyclic if its Laplacian spectrum is not completely real. The essential cyclicity implies the presence of directed cycles, but not vice versa. The problem of characterizing essential cyclicity in terms of graph topology is difficult and yet unsolved. Its solution is important for some applications of graph theory, including that in decentralized control. In the present paper, this problem is solved with respect to the class of digraphs with ring structure, which models some typical communication networks. It is shown that the digraphs in this class are essentially cyclic, except for certain specified digraphs. The main technical tool we employ is the Chebyshev polynomials of the second kind. A by-product of this study is a theorem on the zeros of polynomials that differ by one from the products of Chebyshev polynomials of the second kind. We also consider the problem of essential cyclicity for weighted digraphs and enumerate the spanning trees in some digraphs with ring structure.Comment: 19 pages, 8 figures, Advances in Applied Mathematics: accepted for publication (2010) http://dx.doi.org/10.1016/j.aam.2010.01.00