Analysis of Fisher Information and the Cram\'{e}r-Rao Bound for Nonlinear Parameter Estimation after Compressed Sensing

TL;DR

This paper investigates how compressed sensing with complex random matrices affects Fisher information and the Cramér-Rao Bound in parameter estimation, providing distributions to quantify information loss and guide compression ratio choices.

Contribution

It derives the distribution of Fisher information and CRB after compression with right-orthogonally invariant matrices, advancing understanding of information loss in compressed sensing.

Findings

Fisher information matrix follows a complex matrix beta distribution after compression

Distribution of CRB is derived for compressed sensing scenarios

Guidelines for selecting compression ratios based on CRB loss are provided

Abstract

In this paper, we analyze the impact of compressed sensing with complex random matrices on Fisher information and the Cram\'{e}r-Rao Bound (CRB) for estimating unknown parameters in the mean value function of a complex multivariate normal distribution. We consider the class of random compression matrices whose distribution is right-orthogonally invariant. The compression matrix whose elements are i.i.d. standard normal random variables is one such matrix. We show that for all such compression matrices, the Fisher information matrix has a complex matrix beta distribution. We also derive the distribution of CRB. These distributions can be used to quantify the loss in CRB as a function of the Fisher information of the non-compressed data. In our numerical examples, we consider a direction of arrival estimation problem and discuss the use of these distributions as guidelines for choosing compression ratios based on the resulting loss in CRB.