Boyer, Rémy, Babadi, Behtash, Kalouptsidis, Nicholas, Tarokh, Vahid, Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Department of Electrical and Computer Engineering [Univ. of Maryland] (ECE - University of Maryland), National and Kapodistrian University of Athens (NKUA), and Harvard University [Cambridge]
International audience; Given N noisy measurements denoted by y and an overcom-plete Gaussian dictionary, A, the authors in [1] establish the existence and the asymptotic statistical efficiency of an unbiased estimator unaware of the locations of the non-zero entries, collected in set I, in the deterministic L-sparse signal x. More precisely, there exists an estimator x(y, A) unaware of set I with a variance reaching the oracle-CRB (Cramér-Rao Bound) in the doubly asymptotic scenario, i.e., for N, L → ∞ and L/N → α ∈ (0, 1). As was noted in [2] the result remains true even though the proposed closed-form expression of the variance of the estimator x(y, A) is incorrect. In this note, we correct this expression by providing an explicit formula and discuss its practical usefulness. Finally, the new expression allows to correct the misleading comprehension of the sparse signal estimation performance suggested in [1]. I. MAIN RESULT OF [1] Let y be the N × 1 noisy measurement vector given by y = Ax + n where A is a non-stochastic N × M matrix with controlled growing dimensions according to limN,L→∞ L/N = α ∈ (0, 1). An entry of matrix A is generated as a single realization of an i.i.d. Normal distribution N (0, 1), x is a deterministic L-sparse vector on set I and n is a centered circular white Gaussian noise of variance σ 2. The definition of the oracle (doubly) asymptotic CRB is given hereafter. Definition 1.1: The oracle-CRB in the doubly asymptotic scenario is defined according to C