Fast Graph Filters for Decentralized Subspace Projection
Abstract
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 minimumorder 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 alternatingdirection method of multipliers. An exhaustive simulation study demonstrates that the obtained filters can effectively obtain subspace projections markedly faster than existing algorithms.
 Publication:

arXiv eprints
 Pub Date:
 November 2020
 DOI:
 10.48550/arXiv.2011.07579
 arXiv:
 arXiv:2011.07579
 Bibcode:
 2020arXiv201107579R
 Keywords:

 Statistics  Computation