On the Achievability of Cram\'er-Rao Bound In Noisy Compressed Sensing

TL;DR

This paper extends previous results on noisy compressed sensing, showing that the Cramer-Rao bound can be achieved with a typical estimator even when the measurement matrix is deterministic, under certain concentration conditions.

Contribution

It generalizes the achievability of the Cramer-Rao bound to deterministic measurement matrices satisfying concentration inequalities, removing the Gaussianity assumption.

Findings

Cramer-Rao bound is achievable with deterministic matrices under concentration conditions.

The results extend previous Gaussian-based assumptions to broader classes of matrices.

The typical estimator remains effective beyond Gaussian measurement matrices.

Abstract

Recently, it has been proved in Babadi et al. that in noisy compressed sensing, a joint typical estimator can asymptotically achieve the Cramer-Rao lower bound of the problem.To prove this result, this paper used a lemma,which is provided in Akcakaya et al,that comprises the main building block of the proof. This lemma is based on the assumption of Gaussianity of the measurement matrix and its randomness in the domain of noise. In this correspondence, we generalize the results obtained in Babadi et al by dropping the Gaussianity assumption on the measurement matrix. In fact, by considering the measurement matrix as a deterministic matrix in our analysis, we find a theorem similar to the main theorem of Babadi et al for a family of randomly generated (but deterministic in the noise domain) measurement matrices that satisfy a generalized condition known as The Concentration of Measures Inequality. By this, we finally show that under our generalized assumptions, the Cramer-Rao bound of the estimation is achievable by using the typical estimator introduced in Babadi et al.