Fast Graph Filters for Decentralized Subspace Projection

TL;DR

This paper introduces fast, decentralized graph filter methods for sensor networks to perform subspace projections efficiently, reducing iteration count and improving speed over existing approaches.

Contribution

It develops novel graph filter design techniques using convex relaxation and optimization algorithms to achieve near-minimal filter order and faster convergence in decentralized settings.

Findings

Filters significantly reduce the number of iterations needed for subspace projection.

Proposed algorithms outperform existing methods in simulation studies.

Effective approximation of projections with trade-offs between speed and accuracy.

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 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.