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Canonical correlation analysis (CCA) is a widely used data analysis tool that allows to assess the correlation between two distinct sets of signals. It computes optimal linear combinations of the signals in both sets such that the resulting signals are maximally correlated. The weight vectors defining these optimal linear combinations are referred to as “principal CCA directions”. In addition to this particular type of data analysis, CCA is also often used as a blind source separation (BSS) technique, i.e., under certain assumptions, the principal CCA directions have certain demixing properties. In this paper, we propose a distributed CCA (DCCA) algorithm that can operate in wireless sensor networks (WSNs) with a fully connected or a tree topology. The algorithm estimates the Q principal CCA directions from the sensor signal observations collected by the different nodes in the WSN and extracts the corresponding sources. These network-wide principal CCA directions are estimated in a time-recursive fashion without explicitly constructing the corresponding network-wide correlation matrices, i.e., without the need for data centralization. Instead, each node locally computes smaller CCA problems and only transmits compressed sensor signal observations (of dimension Q), which significantly reduces the bit rate over the wireless links of the WSN. We prove convergence and optimality of the DCCA algorithm, and we demonstrate its performance by means of numerical simulations in a blind source separation scenario. [ABSTRACT FROM AUTHOR]

The generalized eigenvalue decomposition (GEVD) of a pair of matrices generalizes the concept of the eigenvalue decomposition (EVD) of a single matrix. It is a widely used tool in signal processing applications, in particular in a context of spatial filtering and subspace estimation. In this paper, we describe a distributed adaptive algorithm to estimate generalized eigenvectors (GEVCs) of a pair of sensor signal covariance matrices in a fully connected wireless sensor network. The algorithm computes these GEVCs in an iterative fashion without explicitly constructing the full network-wide covariance matrices. Instead, the nodes only exchange compressed sensor signal observations, providing a significant reduction in per-node communication and computational cost compared to the scenario where all the raw sensor signal observations are collected and processed in a fusion center. [ABSTRACT FROM AUTHOR]

Abstract: We describe a distributed adaptive algorithm to estimate the eigenvectors corresponding to the Q largest or smallest eigenvalues of the network-wide sensor signal covariance matrix in a wireless sensor network. The proposed algorithm recursively updates the eigenvector estimates without explicitly constructing the full covariance matrix that defines them, i.e., without centralizing all the raw sensor signal observations. By only sharing fused Q-dimensional observations, each node obtains estimates of (a) the node-specific entries of the Q covariance matrix eigenvectors, and (b) Q-dimensional projections of the full set of sensor signal observations onto the Q eigenvectors. We also explain how the latter can be used for, e.g., compression and reconstruction of the sensor signal observations based on principal component analysis (PCA), in which each node acts as a data sink. We describe a version of the algorithm for fully-connected networks, as well as for partially-connected networks. In the latter case, we assume that the network has been pruned to a tree topology to avoid cycles in the network. We provide convergence proofs, as well as numerical simulations to demonstrate the convergence and optimality of the algorithm. [Copyright &y& Elsevier]

Abstract: The Fiedler vector of a graph is the eigenvector corresponding to the smallest non-trivial eigenvalue of the graph''s Laplacian matrix. The entries of the Fiedler vector are known to provide a powerful heuristic for topology inference, e.g., to identify densely connected node clusters, to search for bottleneck links in the information dissemination, or to increase the overall connectivity of the network. In this paper, we consider ad hoc networks where the nodes can process and exchange data in a synchronous fashion, and we propose a distributed algorithm for in-network estimation of the Fiedler vector and the algebraic connectivity of the corresponding network graph. The algorithm is fully scalable with respect to the network size in terms of per-node computational complexity and data transmission. Simulation results demonstrate the performance of the algorithm. [Copyright &y& Elsevier]

Total least squares (TLS) estimation is a popular solution technique for overdetermined systems of linear equations with a noisy data matrix. In this paper, we revisit the distributed total least squares (D-TLS) algorithm, which operates in an ad hoc network, where each node has access to a subset of the linear equations. The D-TLS algorithm computes the TLS solution of the full system of equations in a fully distributed fashion (without fusion center). To reduce the large computational complexity due to an eigenvalue decomposition (EVD) at every node and in each iteration, we modify the D-TLS algorithm based on inverse power iterations (IPIs). In each step of the modified algorithm, a single IPI is performed, which significantly reduces the computational complexity. We show that this IPI-based D-TLS algorithm still converges to the network-wide TLS solution under certain assumptions, which are often satisfied in practice. We provide simulation results to demonstrate the convergence of the algorithm, even when some of these assumptions are not satisfied. [ABSTRACT FROM PUBLISHER]

In this paper, we consider the linearly constrained distributed adaptive node-specific signal estimation (LC-DANSE) algorithm, which generates a node-specific linearly constrained minimum variance (LCMV) beamformer, i.e., with node-specific linear constraints, at each node of a wireless sensor network. The algorithm significantly reduces the number of signals that are exchanged between nodes, and yet obtains the optimal LCMV beamformers as if each node has access to all the signals in the network. We consider the case where all the steering vectors are known, as well as the blind beamforming case where the steering vectors are not known. We formally prove convergence and optimality for both versions of the LC-DANSE algorithm. We also consider the case where nodes update their local beamformers simultaneously instead of sequentially, and we demonstrate by means of simulations that applying a relaxation is often required to obtain a converging algorithm in this case. We also provide simulation results that demonstrate the effectiveness of the algorithm in a realistic speech enhancement scenario. [ABSTRACT FROM PUBLISHER]

We study the problem of distributed least-squares estimation over ad hoc adaptive networks, where the nodes have a common objective to estimate and track a parameter vector. We consider the case where there is stationary additive colored noise on both the regressors and the output response, which results in biased local least-squares estimators. Assuming that the noise covariance can be estimated (or is known a priori), we first propose a bias-compensated recursive least-squares algorithm (BC-RLS). However, this bias compensation increases the variance or the mean-square deviation (MSD) of the local estimators, and errors in the noise covariance estimates may still result in residual bias. We demonstrate that the MSD and residual bias can then be significantly reduced by applying diffusion adaptation, i.e., by letting nodes combine their local estimates with those of their neighbors. We derive a necessary and sufficient condition for mean-square stability of the algorithm, under some mild assumptions. Furthermore, we derive closed-form expressions for its steady-state mean and mean-square performance. Simulation results are provided, which agree well with the theoretical results. We also consider some special cases where the mean-square stability improvement of diffusion BC-RLS over BC-RLS can be mathematically verified. [ABSTRACT FROM AUTHOR]

Total least squares (TLS) is a popular solution technique for overdetermined systems of linear equations, where both the right-hand side and the input data matrix are assumed to be noisy. We consider a TLS problem in an ad hoc wireless sensor network, where each node collects observations that yield a node-specific subset of linear equations. The goal is to compute the TLS solution of the full set of equations in a distributed fashion, without gathering all these equations in a fusion center. To facilitate the use of the dual-based subgradient algorithm (DBSA), we transform the TLS problem to an equivalent convex semidefinite program (SDP), based on semidefinite relaxation (SDR). This allows us to derive a distributed TLS (D-TLS) algorithm, that satisfies the conditions for convergence of the DBSA, and obtains the same solution as the original (unrelaxed) TLS problem. Even though we make a detour through SDR and SDP theory, the resulting D-TLS algorithm relies on solving local TLS-like problems at each node, rather than computationally expensive SDP optimization techniques. The algorithm is flexible and fully distributed, i.e., it does not make any assumptions on the network topology and nodes only share data with their neighbors through local broadcasts. Due to the flexibility and the uniformity of the network, there is no single point of failure, which makes the algorithm robust to sensor failures. Monte Carlo simulation results are provided to demonstrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

We present a distributed adaptive node-specific signal estimation (DANSE) algorithm that operates in a wireless sensor network with a tree topology. The algorithm extends the DANSE algorithm for fully connected sensor networks, as described in previous work. It is argued why a tree topology is the natural choice if the network is not fully connected. If the node-specific desired signals share a common latent signal subspace, it is shown that the distributed algorithm converges to the same linear MMSE solutions as obtained with the centralized version of the algorithm. The computational load is then shared between the different nodes in the network, and nodes exchange only linear combinations of their sensor signal observations and data received from their neighbors. Despite the low connectivity of the network and the multi-hop signal paths, the algorithm is fully scalable in terms of communication bandwidth and computational power. Two different cases are considered concerning the communication protocol between the nodes: point-to-point transmission and local broadcasting. The former assumes that there is a reserved communication link between node-pairs, whereas with the latter, nodes communicate the same data to all of their neighbors simultaneously. The convergence properties of the algorithm are demonstrated by means of numerical examples. [ABSTRACT FROM AUTHOR]

• Fusion rules for distributed adaptive node-specific signal estimation (DANSE) are not optimal when communication links are noisy in a wireless sensor network. • Fusion rules which take noisy links into account are developed, resulting in new algorithm named N-DANSE. • The strategy to prove convergence of N-DANSE is different from the strategy used for DANSE without noisy links, and includes the latter as a special case. • N-DANSE performs better than DANSE in scenarios with noisy links. • N-DANSE is consistently closer to the optimal performance than DANSE. We consider a distributed signal estimation problem in a wireless sensor network where each node aims to estimate a node-specific desired signal using all sensor signals available in the network. In this setting, the distributed adaptive node-specific signal estimation (DANSE) algorithm is able to learn optimal fusion rules with which the nodes fuse their sensor signals, as the fused signals are then transmitted between the nodes. Under the assumption of transmission without errors, DANSE achieves the performance of centralized estimation. However, noisy communication links introduce errors in these transmitted signals, e.g., due to quantization or communication errors. In this paper we show fusion rules which take additive noise in the transmitted signals into account at almost no increase in computational complexity, resulting in a new algorithm denoted as 'noisy-DANSE' (N-DANSE). As the convergence proof for DANSE cannot be straightforwardly generalized to the case with noisy links, we use a different strategy to prove convergence of N-DANSE, which also proves convergence of DANSE without noisy links as a special case. We validate the convergence of N-DANSE and compare its performance with the original DANSE through numerical simulations, which demonstrate the superiority of N-DANSE over the original DANSE in noisy links scenarios. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of distributed adaptive estimation of node-specific signals for signal enhancement or noise reduction in wireless sensor networks with multi-sensor nodes. The estimation is performed by a multi-channel Wiener filter (MWF) in which a low-rank approximation based on a generalized eigenvalue decomposition (GEVD) is incorporated. In non-stationary or low-SNR conditions, this GEVD-based MWF has been demonstrated to be more robust than the original MWF. In a centralized realization where a fusion center has access to all the nodes’ sensor signal observations, the network-wide sensor signal correlation matrices and the low-rank approximation can be directly estimated and used to compute the network-wide GEVD-based MWF. However, in this paper, we aim to avoid centralizing the sensor signal observations, in which case the network-wide sensor signal correlation matrices cannot be estimated. To this end, we start from the so-called distributed adaptive node-specific signal estimation (DANSE) algorithm, and include GEVD-based low-rank approximations in the per-node local computations. Remarkably, the new algorithm is able to significantly compress the signal observations transmitted between the nodes, while still converging to the network-wide GEVD-based MWF as if each node would have access to all sensor signal observations, even though the low-rank approximations are applied locally at each node. We provide a theoretical convergence analysis, which shows that the algorithm converges to the network-wide GEVD-based MWF under conditions that are less strict than in the original DANSE algorithm. The convergence and performance of the algorithm are further investigated via numerical simulations. [ABSTRACT FROM AUTHOR]

In multiple speaker scenarios, the linearly constrained minimum variance (LCMV) beamformer is a popular microphone array-based speech enhancement technique, as it allows minimizing the noise power while maintaining a set of desired responses towards different speakers. Here, we address the algorithmic challenges arising when applying the LCMV beamformer in wireless acoustic sensor networks (WASNs), which are a next-generation technology for audio acquisition and processing. We review three optimal distributed LCMV-based algorithms, which compute a network-wide LCMV beamformer output at each node without centralizing the microphone signals. Optimality here refers to equivalence to a centralized realization where a single processor has access to all signals. We derive and motivate the algorithms in an accessible top-down framework that reveals their underlying relations. We explain how their differences result from their different design criterion (node-specific versus common constraints sets), and their different priorities for communication bandwidth, computational power, and adaptivity. Furthermore, although originally proposed for a fully connected WASN, we also explain how to extend the reviewed algorithms to the case of a partially connected WASN, which is assumed to be pruned to a tree topology. Finally, we discuss the advantages and disadvantages of the various algorithms [ABSTRACT FROM AUTHOR]

In this paper, we consider cooperative node-specific direction-of-arrival (DOA) estimation in a fully connected wireless acoustic sensor network (WASN). We consider a scenario where each node is equipped with a local microphone array with a known geometry, but where the position of the nodes, as well as their relative geometry and hence the between-nodes signal coherence model is unknown. The local array geometry in each node defines node-specific DOAs with respect to a set of target speech sources and the aim is to estimate these in each node. We assume a noisy environment with localized and/or diffuse noise sources, i.e., the noise can be correlated over the different microphones. A distributed noise reduction algorithm can then be applied as a preprocessing step to denoise all the microphone signals of the WASN, based on the distributed adaptive node-specific signal estimation (DANSE) algorithm. The denoised local microphone signals can then be used in each node to estimate the node-specific DOAs by using a subspace-based DOA estimation, involving a (generalized) eigenvalue decomposition of the local microphone signal correlation matrices. It is seen that the fused microphone signals that are exchanged between the nodes in the DANSE algorithm can also be included in these correlation matrices to obtain improved DOA estimates, leading to a cooperative integrated noise reduction and DOA estimation scheme, where the noise reduction can actually be shortcut. The improved performance achieved by this cooperative DOA estimation is demonstrated by means of numerical simulations for two different subspace-based DOA estimation methods (MUSIC and ESPRIT). [ABSTRACT FROM AUTHOR]

Diffusion adaptation techniques have shown great promise in addressing the problem of node-specific distributed estimation where the nodes in the network are interested in different, possibly overlapping, sets of parameters. In this work, node-specific distributed detection, which has remained largely unexamined, is considered. In particular, the problem is formulated as one of binary hypothesis testing at each node for each of its parameters of interest. A distributed, online solution for this problem is sought based on diffusion adaptation techniques. In this setting, a signal to be detected by one node constitutes interference that may compromise the ability of the other nodes to detect their signals of interest reliably. Under mild assumptions on the data and network, it is shown that, for sufficiently small adaptation step-sizes, interference can be kept in check. Local detectors are developed where the test-statistics and thresholds adapt to changing conditions in real time. The distributed algorithm is analyzed; and its detection performance characterized and illustrated through numerical simulations. [ABSTRACT FROM AUTHOR]

Abstract: A wireless sensor network is envisaged that performs signal estimation by means of the distributed adaptive node-specific signal estimation (DANSE) algorithm. This wireless sensor network has constraints such that only a subset of the nodes are used for the estimation of a signal. While an optimal node selection strategy is NP-hard due to its combinatorial nature, we propose a greedy procedure that can add or remove nodes in an iterative fashion until the constraints are satisfied based on their utility. With the proposed definition of utility, a centralized algorithm can efficiently compute each nodes's utility at hardly any additional computational cost. Unfortunately, in a distributed scenario this approach becomes intractable. However, by using the convergence and optimality properties of the DANSE algorithm, it is shown that for node removal, each node can efficiently compute a utility upper bound such that the MMSE increase after removal will never exceed this value. In the case of node addition, each node can determine a utility lower bound such that the MMSE decrease will always exceed this value once added. The greedy node selection procedure can then use these upper and lower bounds to facilitate distributed node selection. [Copyright &y& Elsevier]

Wireless sensor networks are often deployed over a large area of interest and therefore the quality of the sensor signals may vary significantly across the different sensors. In this case, it is useful to have a measure for the importance or the so-called “utility” of each sensor, e.g., for sensor subset selection, resource allocation or topology selection. In this paper, we consider the efficient calculation of sensor utility measures for four different signal estimation or beamforming algorithms in an adaptive context. We use the definition of sensor utility as the increase in cost (e.g., mean-squared error) when the sensor is removed from the estimation procedure. Since each possible sensor removal corresponds to a new estimation problem (involving less sensors), calculating the sensor utilities would require a continuous updating of K different signal estimators (where K is the number of sensors), increasing computational complexity and memory usage by a factor K. However, we derive formulas to efficiently calculate all sensor utilities with hardly any increase in memory usage and computational complexity compared to the signal estimation algorithm already in place. When applied in adaptive signal estimation algorithms, this allows for on-line tracking of all the sensor utilities at almost no additional cost. Furthermore, we derive efficient formulas for sensor removal, i.e., for updating the signal estimator coefficients when a sensor is removed, e.g., due to a failure in the wireless link or when its utility is too low. We provide a complexity evaluation of the derived formulas, and demonstrate the significant reduction in computational complexity compared to straightforward implementations. [ABSTRACT FROM PUBLISHER]