A significant number of linear inference problems in wireless sensor networks can be solved by projecting the observed signal onto a given subspace. Decentralized approaches avoid the need for performing such an operation at a central processor, thus, reducing the congestion and increasing the robustness of the communication network. Unfortunately, existing decentralized approaches either confine themselves to a reduced family of subspace projection tasks or need an infinite number of iterations to obtain the exact projection. To remedy these limitations, this paper develops a framework for computing a wide class of subspace projections in a decentralized fashion by relying on the notion of graph filtering. To this end, a methodology to obtain the shift matrix and the corresponding filter coefficients that provide exact subspace projection in a nearly minimal number of iterations is proposed. Numerical experiments corroborate the merits of the proposed approach.

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In this paper, we consider the problem of improving the convergence speed of an average consensus gossip algorithm by sparsifying a sufficiently dense network graph. Thus, instead of adding links, as usually proposed in the literature, or globally optimizing the mixing matrix of the gossip algorithm for a given network, which requires global knowledge at every node, we find a sparser network that has better spectral properties and faster convergence than the original denser one. This allows to reduce simultaneously both the convergence time and the communication cost involved in the execution of the gossip algorithm. We first show why it is possible to sparsify a network while increasing its convergence rate and also that there exists an optimal fraction of links to be removed. As a benchmark, we devise a centralized method that selects in an optimal way the set of links to be removed from the original network. Then, we propose a low complexity and scalable decentralized protocol requiring only local information at each node, which also generates a sparser network having a substantially better convergence rate. Simulation results are presented to verify and show clearly the efficiency of our approach.

A number of inference problems with sensor networks involve projecting a measured signal onto a given subspace. In existing decentralized approaches, sensors communicate with their local neighbors to obtain a sequence of iterates that asymptotically converges to the desired projection. In contrast, the present paper develops methods that produce these projections in a finite and approximately minimal number of iterations. Building upon tools from graph signal processing, the problem is cast as the design of a graph filter which, in turn, is reduced to the design of a suitable graph shift operator. Exploiting the eigenstructure of the projection and shift matrices leads to an objective whose minimization yields approximately minimum-order graph filters. To cope with the fact that this problem is not convex, the present work introduces a novel convex relaxation of the number of distinct eigenvalues of a matrix based on the nuclear norm of a Kronecker difference. To tackle the case where there exists no graph filter capable of implementing a certain subspace projection with a given network topology, a second optimization criterion is presented to approximate the desired projection while trading the number of iterations for approximation error. Two algorithms are proposed to optimize the aforementioned criteria based on the alternating-direction method of multipliers. An exhaustive simulation study demonstrates that the obtained filters can effectively obtain subspace projections markedly faster than existing algorithms.