TY - JOUR

T1 - Non-Bayesian Estimation Framework for Signal Recovery on Graphs

AU - Routtenberg, Tirza

N1 - Funding Information:
Manuscript received May 5, 2020; revised September 9, 2020 and December 28, 2020; accepted January 23, 2021. Date of publication January 29, 2021; date of current version February 19, 2021. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Yao Xie. This work was supported by The Israel Science Foundation under Grant 1173/16.
Publisher Copyright:
© 1991-2012 IEEE.

PY - 2021/1/1

Y1 - 2021/1/1

N2 - Graph signals arise from physical networks, such as power and communication systems, or as a result of a convenient representation of data with complex structure, such as social networks. We consider the problem of general graph signal recovery from noisy, corrupted, or incomplete measurements and under structural parametric constraints, such as smoothness in the graph frequency domain. In this paper, we formulate the graph signal recovery as a non-Bayesian estimation problem under a weighted mean-squared-error (WMSE) criterion, which is based on a quadratic form of the Laplacian matrix of the graph, and its trace WMSE is the Dirichlet energy of the estimation error w.r.t. the graph. The Laplacian-based WMSE penalizes estimation errors according to their graph spectral content and is a difference-based cost function which accounts for the fact that in many cases signal recovery on graphs can only be achieved up to a constant addend. We develop a new Cramér-Rao bound (CRB) on the Laplacian-based WMSE and present the associated Lehmann unbiasedness condition w.r.t. the graph. We discuss the graph CRB and estimation methods for the fundamental problems of 1) a linear Gaussian model with relative measurements; and 2) bandlimited graph signal recovery. We develop sampling allocation policies that optimize sensor locations in a network for these problems based on the proposed graph CRB. Numerical simulations on random graphs and on electrical network data are used to validate the performance of the graph CRB and sampling policies.

AB - Graph signals arise from physical networks, such as power and communication systems, or as a result of a convenient representation of data with complex structure, such as social networks. We consider the problem of general graph signal recovery from noisy, corrupted, or incomplete measurements and under structural parametric constraints, such as smoothness in the graph frequency domain. In this paper, we formulate the graph signal recovery as a non-Bayesian estimation problem under a weighted mean-squared-error (WMSE) criterion, which is based on a quadratic form of the Laplacian matrix of the graph, and its trace WMSE is the Dirichlet energy of the estimation error w.r.t. the graph. The Laplacian-based WMSE penalizes estimation errors according to their graph spectral content and is a difference-based cost function which accounts for the fact that in many cases signal recovery on graphs can only be achieved up to a constant addend. We develop a new Cramér-Rao bound (CRB) on the Laplacian-based WMSE and present the associated Lehmann unbiasedness condition w.r.t. the graph. We discuss the graph CRB and estimation methods for the fundamental problems of 1) a linear Gaussian model with relative measurements; and 2) bandlimited graph signal recovery. We develop sampling allocation policies that optimize sensor locations in a network for these problems based on the proposed graph CRB. Numerical simulations on random graphs and on electrical network data are used to validate the performance of the graph CRB and sampling policies.

KW - Dirichlet energy

KW - Laplacian matrix

KW - Non-Bayesian parameter estimation

KW - graph Cramér-Rao bound

KW - graph signal processing

KW - graph signal recovery

KW - sensor placement

UR - http://www.scopus.com/inward/record.url?scp=85100494295&partnerID=8YFLogxK

U2 - 10.1109/TSP.2021.3054995

DO - 10.1109/TSP.2021.3054995

M3 - Article

AN - SCOPUS:85100494295

SN - 1053-587X

VL - 69

SP - 1169

EP - 1184

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

M1 - 9340602

ER -