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 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.

Robust Adaptive Estimation of Graph Signals Based on Welsch Loss

“…The area of graph signal processing (GSP) has received extensive attention [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Graph signal processing has been found in social and economic networks, climate analysis, traffic patterns, marketing preferences, and so on [19][20][21][22][23][24][25][26].…”

Comments on "Design of Asymmetric Shift Operators for Efficient Decentralized Subspace Projection"

“…This correspondence comments on [1], where the goal is to design shift matrices for graph filters whose output is the result of projecting their input onto a given subspace. The paper relies on a Schur decomposition S = W (D + Q)W ⊤ of the sought shift matrix S. The goal is therefore the same as in the earlier paper [2] with the exception that S is required to be symmetric in [2]. The reason for such a requirement in [2] was to render the problem tractable.…”

“…The paper relies on a Schur decomposition S = W (D + Q)W ⊤ of the sought shift matrix S. The goal is therefore the same as in the earlier paper [2] with the exception that S is required to be symmetric in [2]. The reason for such a requirement in [2] was to render the problem tractable. The work in [1] claims to solve the problem when that assumption is lifted.…”

“…Finally, although not mentioned in [1], the technique to deal with constraint (11f) therein was proposed in [2].…”

“…Theorem 2. Although not mentioned, this theorem is based on Theorem 2 and Corollary 1 in [2]; see also the explanation afterwards. In fact, Appendix C in [1] is an adaptation of [2, Appendix C], including the same steps and equations up to small modifications.…”