Improved least-squares-based combiners for diffusion networks Jesus Fernandez-Bes ∗ , Luis A. Azpicueta-Ruiz ∗ , Magno T. M. Silva † , and Jer´ onimo Arenas-Garc´ ıa ∗ ∗ Universidad Carlos III de Madrid † Universidade de S˜ ao Paulo Legan´ es, 28911 Spain S˜ ao Paulo 05508-010, Brazil {jesusfbes,lazpicueta,jarenas}@tsc.uc3m.es magno@lps.usp.br Abstract—Adaptive networks have received great attention during recent years. In diffusion strategies, nodes diffuse their estimations to neighbors, and construct improved estimates by combining all information received by other nodes. When nodes work in heterogeneous conditions, it is reasonable to assign com- bination weights that take into account the performance of each node; thus, different schemes that implement adaptive combiners have been recently proposed. In this paper, we propose a novel scheme for adaptive combiners which attempts to minimize a least-squares cost function. The novelty in our proposal relies on making the adaptive combiners convex, by projection onto the standard simplex, what result in a numerically more stable implementation. The convergence and steady-state properties of the new scheme are analyzed theoretically, and its performance is experimentally evaluated with respect to existing methods. I. I NTRODUCTION AND PROBLEM FORMULATION Recently, adaptive networks have gained considerable at- tention as an efficient way of estimating certain parameters of interest using the information from data collected by sensor nodes distributed over a region (see, e.g., [1], [2] and their references). In many applications, these networks must track the variations in the data statistics, which justifies the need for adaptiveness. In this context, different distributed estima- tion strategies can be considered. In particular, in diffusion networks, nodes diffuse their estimates to the network, so that each node can combine its own estimation with those received from neighboring nodes. Fig. 1 shows a diffusion network with N nodes, where node k combines the estimations received from neighbors ¯ N k = {1, 2,r} with its own estimation. Each node takes a measurement {d k (n), u k (n)} at each time instant n, where d k (n) represents a desired signal and u k (n) is a length-M input regressor vector, related through the usual linear regression model d k (n)= u T k (n)w o (n − 1) + v k (n). Here, v k (n) is measurement noise while w o (n) is a length-M column parameter vector. Note that w o (n) is assumed to be common to all nodes, and the network goal is its estimation. Different approaches can be followed in order to obtain the combined estimation at each node. Recently, we have proposed a method [3] where each node performs its estimation in two steps: (i) it adapts and preserves its local estimation ψ k (n) and (ii) combines it with the combined estimates received from the The work of Fernandez-Bes, Azpicueta-Ruiz, and Arenas-Garc´ ıa was partly supported by MINECO projects TEC2011-22480 and PRI-PIBIN-2011-1266. The work of Fernandez-Bes was also supported by Spanish MECD FPU program. The work of Silva was partly supported by CNPq under Grant 302423/2011-7 and FAPESP under Grant 2012/24835-1. N k k N 1 2 r 3 {d k (n), u k (n)} {d 1 (n), u 1 (n)} {d 2 (n), u 2 (n)} {d r (n), u r (n)} {d N (n), u N (n)} {d 3 (n), u 3 (n)} Fig. 1. Diffusion network with N nodes: at time n, every node k takes a measurement {d k (n), u k (n)}. The neighborhood of node k in this network is N k = {1, 2, r, k}, and its cardinality is denoted as N k =4. neighboring nodes at the previous iteration w ℓ (n−1). Thus, its combined estimation, that will be transmitted to its neighbors diffusing the information, is given by w k (n)= c kk (n)ψ k (n)+  ℓ∈ ¯ N k c ℓk (n)w ℓ (n − 1), (1) where ¯ N k represents the neighborhood of node k excluding itself and c ℓk (n), for k = 1, 2,...,N and ℓ ∈N k , are the combination weights assigned by node k to the different estimates combined in (1), with N k being the neighborhood of node k including itself. It should be evident that, with an adequate selection of c ℓk (n), w k (n) can potentially provide an improved estimate of w o (n) with respect to ψ k (n). We should notice that most papers on diffusion networks assume fixed combination weights, whose values are computed based on the network topology only. However, these static combination rules do not take into account diversity among nodes, or the fact that these may be operating under different signal-to-noise ratio (SNR) conditions, resulting in suboptimal performance when the SNR varies across the network. For this reason, some schemes that implement adaptive combination weights have been recently proposed in [1], [2]. Although these adaptive solutions improve the performance of the networks when compared to static combiners, the learned combination parameters may still be suboptimal during the convergence or when tracking time-varying solutions, espe- cially when different step sizes are used across the network nodes. This occurs due to the fact that some of the assumptions used in the derivations of these adaptive rules hold mainly in stationary scenarios and steady-state conditions. Our proposal in [3], [4] updates the combiners using a least-squares (LS) rule. Compared to the solutions of [1],