On the MMSE Estimation of Norm of a Gaussian Vector under Additive White Gaussian Noise with Randomly Missing Input Entries

TL;DR

This paper analyzes the MMSE estimation of the Euclidean norm of a Gaussian vector with missing entries under AWGN, deriving asymptotic and noise variance-dependent MSE expressions, extending prior work to missing data scenarios.

Contribution

It introduces the MMSE estimator for the Gaussian vector norm with missing data and noise, providing new theoretical MSE formulas and asymptotic behavior analysis.

Findings

MSE normalized by n tends to 0 as n→∞ with fixed K/n ratio.

Explicit MSE expressions derived for noise variance tending to 0 or ∞.

Extension of previous results to scenarios with missing data entries.

Abstract

This paper considers the task of estimating the $l_2$ norm of a $n$-dimensional random Gaussian vector from noisy measurements taken after many of the entries of the vector are \emph{missed} and only $K\ (0\le K\le n)$ entries are retained and others are set to $0$. Specifically, we evaluate the minimum mean square error (MMSE) estimator of the $l_2$ norm of the unknown Gaussian vector performing measurements under additive white Gaussian noise (AWGN) on the vector after the data missing and derive expressions for the corresponding mean square error (MSE). We find that the corresponding MSE normalized by $n$ tends to $0$ as $n\to \infty$ when $K/n$ is kept constant. Furthermore, expressions for the MSE is derived when the variance of the AWGN noise tends to either $0$ or $\infty$. These results generalize the results of Dytso et al.\cite{dytso2019estimating} where the case $K=n$ is considered, i.e. the MMSE estimator of norm of random Gaussian vector is derived from measurements under AWGN noise without considering the data missing phenomenon.