NON-BAYESIAN ESTIMATION FRAMEWORK FOR SIGNAL RECOVERY ON GRAPHS

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 under structural parametric constraints. 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. We develop a new Cram�r-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 network sensor locations 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.