Fast Graph Filters for Decentralized Subspace Projection

Many inference problems with sensor networks involve projecting a measured signal onto a given subspace. In existing decentralized approaches, sensors communicate with their neighbors to obtain a sequence of iterates that asymptotically converges to the desired projection. In contrast, this paper develops methods that produce these projections in a finite and approximately minimal number of iterations. Building upon graph signal processing tools, the problem is cast as a graph filter design 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.