Errata to " Asymptotic Achievability of the Cramér -Rao Bound for Noisy Compressive Sampling "

Remy Boyer (1) , Behtash Babadi (2) , Nicholas Kalouptsidis (3) , Vahid Tarokh (4)

1 L2S - Laboratoire des signaux et systèmes
2 ECE - University of Maryland - Department of Electrical and Computer Engineering [Univ. of Maryland]
3 NKUA - National and Kapodistrian University of Athens
4 Harvard University

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

Domaines: Traitement du signal et de l'image [eess.SP]

Remy Boyer, Behtash Babadi, Nicholas Kalouptsidis, Vahid Tarokh. Errata to " Asymptotic Achievability of the Cramér -Rao Bound for Noisy Compressive Sampling ". IEEE Transactions on Signal Processing, 2017, 65 (18), pp.4973-4974. ⟨10.1109/tsp.2017.2723352⟩. ⟨hal-01299004⟩