Comment on "Asymptotic Achievability of the Cram\'{e}r-Rao Bound for Noisy Compressive Sampling"

TL;DR

This paper clarifies and corrects a previous proof regarding the asymptotic achievability of the Cramér-Rao bound in noisy compressive sampling, providing a more detailed non-asymptotic analysis.

Contribution

It corrects a previous proof by avoiding an erroneous expression and offers a detailed non-asymptotic proof in the context of noisy compressive sensing.

Findings

The original proof remains valid with the correction.

A detailed non-asymptotic proof is provided.

The correction confirms the asymptotic achievability of the Cramér-Rao bound.

Abstract

In [1], we proved the asymptotic achievability of the Cram\'{e}r-Rao bound in the compressive sensing setting in the linear sparsity regime. In the proof, we used an erroneous closed-form expression of $\alpha \sigma^2$ for the genie-aided Cram\'{e}r-Rao bound $\sigma^2 \textrm{Tr} (\mathbf{A}^*_\mathcal{I} \mathbf{A}_\mathcal{I})^{-1}$ from Lemma 3.5, which appears in Eqs. (20) and (29). The proof, however, holds if one avoids replacing $\sigma^2 \textrm{Tr} (\mathbf{A}^*_\mathcal{I} \mathbf{A}_\mathcal{I})^{-1}$ by the expression of Lemma 3.5, and hence the claim of the Main Theorem stands true. In Chapter 2 of the Ph. D. dissertation by Behtash Babadi [2], this error was fixed and a more detailed proof in the non-asymptotic regime was presented. A draft of Chapter 2 of [2] is included in this note, verbatim. We would like to refer the interested reader to the full dissertation, which is electronically archived in the ProQuest database [2], and a draft of which can be accessed through the author's homepage under: http://ece.umd.edu/~behtash/babadi_thesis_2011.pdf.