On the Performance Bound of Sparse Estimation with Sensing Matrix Perturbation

TL;DR

This paper analyzes the fundamental limits of sparse estimation when both the sensing matrix and measurements are corrupted by Gaussian noise, deriving bounds that reveal the impact of matrix perturbation on estimation performance.

Contribution

It derives closed-form expressions for the CCRB and HCRB bounds under sensing matrix perturbation, providing insights into their differences and asymptotic behaviors.

Findings

The CCRB shows a gap between maximal and nonmaximal support cases.

The HCRB eliminates the gap between support cases in the unit sensing matrix scenario.

Numerical results confirm the theoretical bounds and their implications.

Abstract

This paper focusses on the sparse estimation in the situation where both the the sensing matrix and the measurement vector are corrupted by additive Gaussian noises. The performance bound of sparse estimation is analyzed and discussed in depth. Two types of lower bounds, the constrained Cram\'{e}r-Rao bound (CCRB) and the Hammersley-Chapman-Robbins bound (HCRB), are discussed. It is shown that the situation with sensing matrix perturbation is more complex than the one with only measurement noise. For the CCRB, its closed-form expression is deduced. It demonstrates a gap between the maximal and nonmaximal support cases. It is also revealed that a gap lies between the CCRB and the MSE of the oracle pseudoinverse estimator, but it approaches zero asymptotically when the problem dimensions tend to infinity. For a tighter bound, the HCRB, despite of the difficulty in obtaining a simple expression for general sensing matrix, a closed-form expression in the unit sensing matrix case is derived for a qualitative study of the performance bound. It is shown that the gap between the maximal and nonmaximal cases is eliminated for the HCRB. Numerical simulations are performed to verify the theoretical results in this paper.