The decentralized estimation of the sample covariance

Abstract

In this paper we consider the problem of estimating the eigenvectors of the sample covariance matrix of decentralized measurements in a distributed fashion. The need for a distributed scheme is motivated by the many moment based methods that resort to the covariance of the data to extract information from the measurements. For large sensor network, gathering the data at a central processor generates a communication bottleneck. Our algorithm is based on a combination of the so called power method, that is used to compute the eigenvectors, and the average consensus protocol, that is utilized to structure the information exchange into a gossiping protocol. Our work shows how a completely distributed scheme based on near neighbors communications is feasible, and applies the proposed method to the estimation of the direction of arrival of a signal source.

Cloud K-SVD: Computing data-adaptive representations in the cloud

“…However, relatively little attention has been paid to the problem of distributed learning of the geometric structure of data in a data-adaptive manner. Most notable exceptions to this include [7]- [11]. While our work as well as [7]- [11] rely on consensus averaging for computing the underlying geometric structure, we are explicit in our formulation that perfect consensus under arbitrary topologies cannot be achieved.…”

“…The setting studied in some of these works is that data is partitioned horizontally, with each distributed entity responsible for some dimensions of the data [6], [9]. Some of the other works in this direction focus on learning under the assumption of data lying near (linear) subs paces [7], [8], require extensive communications among the distributed entities [10], and ignore some of the technical details associated with processing among distributed entities having interconnections described by graphs of arbi trary topologies [7], [8], [10], [11].…”

Experiments on formation switching for mobile robots

“…This method converges to the maximum eigenvector of C as long as the maximum eigenvalue ofĈ is strictly greater than the other eigenvalues and the vector q(0), an initial random vector, has a non-zero component in the direction of the eigenvector associated to the largest eigenvalue [15]. For simplicity, let us assume thatx = 0.…”

“…Unlike [15], we perform the average consensus protocols for x i (x i · q 1 (n)), i = 1, ..., N , instead of only the inner products and, in our algorithm, each robot does not need to know the number of robots in the team. Finally, we can obtain the estimate of q 1 .…”

“…We estimate the largest eigenvector of the covariance matrix of distributed positions in a decentralized fashion, assuming that the positions are measured by the localization module and the communication links are locally connected. The idea proposed in [15] is based on the decomposition of the so-called power method into a decentralized iterative protocol. The eigenvectors of the covariance matrixĈ can be derived by using a power method as follows…”