Partial estimation of covariance matrices

A classical approach to accurately estimating the covariance matrix Σ of a p-variate normal distribution is to draw a sample of size n > p and form a sample covariance matrix. However, many modern applications operate with much smaller sample sizes, thus calling for estimation guarantees in the regime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \ll p}$$\end{document}. We show that a sample of size n = O(m log6p) is sufficient to accurately estimate in operator norm an arbitrary symmetric part of Σ consisting of m ≤ n nonzero entries per row. This follows from a general result on estimating Hadamard products M · Σ, where M is an arbitrary symmetric matrix.