Diffusion Adaptation Over Networks

Abstract

Adaptive networks are well-suited to perform decentralized information processing and optimization tasks and to model various types of self-organized and complex behavior encountered in nature. Adaptive networks consist of a collection of agents with processing and learning abilities. The agents are linked together through a connection topology, and they cooperate with each other through local interactions to solve distributed optimization, estimation, and inference problems in real-time. The continuous diffusion of information across the network enables agents to adapt their performance in relation to streaming data and network conditions; it also results in improved adaptation and learning performance relative to non-cooperative agents. This article provides an overview of diffusion strategies for adaptation and learning over networks. The article is divided into several sections: 1. Motivation. Comparison Conditions MSD network atc ≤ MSD network cta A doubly stochastic, C right stochastic. MSD network cta,C =I ≤ MSD network cta,C=I C T RvC ≤ Rv, C doubly stochastic, R u,k = Ru, µ k = µ. MSD network cta,C=I ≤ MSD network cta,C =I C T RvC ≥ Rv, C doubly stochastic, R u,k = Ru, µ k = µ. MSD network atc,C =I ≤ MSD network atc,C=I C T RvC ≤ Rv, C doubly stochastic, R u,k = Ru, µ k = µ. MSD network atc,C=I ≤ MSD network atc,C =I C T RvC ≥ Rv, C doubly stochastic, R u,k = Ru, µ k = µ. MSD network atc ≤ MSD network cta ≤ MSD network lms {A, C} doubly stochastic, R u,k = Ru, µ k = µ.

On the Learning Behavior of Adaptive Networks—Part I: Transient Analysis

“…Let A = [a lk ] denote the entries of A. Assumption 1 is automatically satisfied if the product A corresponds to a connected network and there exists at least one a kk > 0 for some node k (i.e., at least one node with a nontrivial self-loop) [21], [23]. It then follows from the Perron-Frobenius Theorem [52] that the matrix A 1 A 0 A 2 has a single eigenvalue at one of multiplicity one and all other eigenvalues are strictly less than one in magnitude, i.e.,…”

Distributed wideband spectrum sensing for cognitive radio networks

“…In addition, given that sensors have unknown different channels, it is hard to compute the PSD of the PU; however, each sensor can adaptively compute an averaged estimate of the received PSD using a diffusion adaptation strategy. Several diffusion adaptation schemes have been proposed in the literature (see [10] and references therein), and one of them is the adapt-then-combine (ATC) diffusion algorithm [11]. It consists of two steps.…”

“…where μ k is a constant positive step size, which is chosen to be sufficiently small to ensure convergence [10]. The second step is a combination stage where the intermediate estimates of the received power spectrum ϕ z,m from the neighborhood of sensor k (z ∈ N k ) are combined through the coefficients {a z,k } to obtain the updated estimatep k,m as followŝ…”

“…More specifically, denoting ξ z,k,m as the power spectrum estimation error (with regard to the true averaged PSD) of node z available at node k, at iteration m, in this paper, following the idea of [10], we use a combination rule where node k combines the local received power spectrum estimates from its neighbors in an inversely proportional manner to the estimation error…”

“…Cognitive users receive the current local PSD estimates of the immediate neighbors and combine them following a specific combination rule. The rule, which is known to all the sensors, can be chosen according to the available information at the sensors, and it can range from a simple averaging operation to a more intelligent weighted combination [10]. In this way, weighting the received power spectrum of neighbors can help those users with bad channels to improve their detection capabilities.…”