The trace ratio optimization (TRO) problem consists of finding an orthonormal basis for the discriminative subspace that maximizes the ratio of two trace operators on two covariance matrices corresponding to two distinctive classes or signal components. The TRO problem is encountered in various signal processing problems such as dimensionality reduction, signal enhancement, and discriminative analysis. In this paper, we propose a distributed and adaptive algorithm for solving the TRO problem in the context of wireless sensor networks (WSNs), where the two matrices involved in the trace ratio operators correspond to the (unknown) spatial correlation of the sensor signals across the nodes in the network. We first focus on fully-connected networks where every node can communicate with each other, but only compressed signals observations can be shared to reduce the communication cost. After showing convergence, we modify the algorithm to operate in WSNs with more general topologies. Simulation results are provided to validate and complement the theoretical results.

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.

No abstract

The distributed adaptive signal fusion (DASF) algorithm is a generic algorithm that can be used to solve various spatial signal and feature fusion optimization problems in a distributed setting such as a wireless sensor network. Examples include principal component analysis, adaptive beamforming, and source separation problems. While the DASF algorithm adaptively learns the relevant second order statistics from the collected sensor data, accuracy problems can arise if the spatial covariance structure of the signals is rapidly changing. In this paper, we propose a method to improve the tracking or convergence speed of the DASF algorithm in a fully-connected sensor network with a broadcast communication protocol. While the improved tracking increases communication cost, we demonstrate that this tradeoff is efficient in the sense that an L-fold increase in bandwidth results in an R times faster convergence with R >> L.

The trace ratio optimization problem consists of maximizing a ratio between two trace operators and often appears in dimensionality reduction problems for denoising or discriminant analysis. In this paper, we propose a distributed and adaptive algorithm to solve the trace ratio optimization problem over network-wide covariance matrices, which capture the spatial correlation across sensors in a wireless sensor network. We focus on fully-connected network topologies, in which case the distributed algorithm reduces the communication bottleneck by only sharing a compressed version of the observed signals at each given node. Despite this compression, the algorithm can be shown to converge to the maximal trace ratio as if all nodes would have access to all signals in the network. We provide simulation results to demonstrate the convergence and optimality properties of the proposed algorithm.