This paper studies the convergence conditions and properties of the distributed adaptive signal fusion (DASF) algorithm, the framework itself having been introduced in a 'Part I' companion paper. The DASF algorithm can be used to solve linear signal and feature fusion optimization problems in a distributed fashion, and is in particular well-suited for solving spatial filtering optimization problems encountered in wireless sensor networks. The convergence conditions and results are provided along with rigorous proofs and analyses, as well as various example problems to which they apply. Additionally, we describe procedures that can be added to the DASF algorithm to ensure convergence in specific cases where some of the technical convergence conditions are not satisfied.

A wireless sensor network (WSN) consists of a collection of sensor nodes, which are equipped with processing and wireless communication facilities to share data between each other. In some WSN applications, it would be relevant for each node to identify which signal components it shares with other nodes in the network. However, this is hard to realize in a distributed context, in particular between node pairs that do not share a direct wireless link. In this paper, we introduce a distributed algorithm for estimating the signal subspace that (on average) is closest to the pairwise intersections between any two of the per-node sensor signal subspaces. In order to facilitate an efficient data fusion, we assume the WSN has (or can be pruned to) a tree-topology. As opposed to a centralized algorithm where all the sensor signals are transmitted to a fusion center (FC), the per-node bandwidth and processing requirements are independent of the network-size and only depend on the number of neighbors per node and a chosen compression parameter. By construction, our algorithm converges to the solution of the so-called "maximum variance" (MAXVAR) formulation of the generalized canonical correlation anlalysis (GCCA) problem in which observations of every node act as a separate "view" in the GCCA formulation. Therefore, even though our work is formalized within a WSN context, it can be used as a generic distributed MAXVAR algorithm in other application contexts as well.

The performance of most array signal processing tasks relies on the presence of correlation between sensor signals.In a wireless sensor network, where sensor nodes are spread out over a relatively large area, it is useful to identify nodes observing similar sensor signals and hence common phenomenons, for example to partition the network according to the observed latent signals and corresponding correlation structure. This can be achieved via the so-called MAXVAR formulation of generalized canonical correlation analysis, which finds a low-dimensional subspace that highlights correlated signal components between multiple nodes' observed signal subspaces. The classical procedure for computing the solutions of MAXVAR consists in performing a generalized eigenvalue decomposition after collecting all the sensors' signals at a fusion center. However, this typically incurs high communication and computational costs. In this paper, we describe a low communication and computational cost distributed algorithm that computes the solutions of MAXVAR without aggregating the nodes' observations at a central location. We show how a subset of those solutions can be used locally by each node to estimate the global correlation structure across all nodes in the network, thereby allowing any node to evaluate the presence of correlated signals at any other node, even if no direct link is shared. We prove the convergence of the algorithm and validate our method for estimating the correlation structure via simulations.

A wireless sensor network often relies on a fusion center to process the data collected by each of its sensing nodes. Such an approach relies on the continuous transmission of raw data to the fusion center, which typically has a major impact on the sensors' battery life. To address this issue in the particular context of spatial filtering and signal fusion problems, we recently proposed the Distributed Adaptive Signal Fusion (DASF) algorithm, which distributively computes a spatial filter expressed as the solution of a smooth optimization problem involving the network-wide sensor signal statistics. In this work, we show that the DASF algorithm can be extended to compute the filters associated with a certain class of non-smooth optimization problems. This extension makes the addition of sparsity-inducing norms to the problem's cost function possible, allowing sensor selection to be performed in a distributed fashion, alongside the filtering task of interest, thereby further reducing the network's energy consumption. We provide a description of the algorithm, prove its convergence, and validate its performance and solution tracking capabilities with numerical experiments.

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