Sparse covariance matrix estimation in high-dimensional deconvolution

TL;DR

This paper introduces spectral thresholding estimators for high-dimensional covariance matrix estimation from noisy data, achieving near-optimal convergence rates under minimal assumptions.

Contribution

It proposes adaptive spectral thresholding methods for covariance estimation in noisy, high-dimensional settings, with theoretical guarantees and practical illustrations.

Findings

Establishes oracle inequalities for the proposed estimators.

Derives minimax convergence rates logarithmic in sample size and dimension.

Demonstrates finite sample performance through numerical examples.

Abstract

We study the estimation of the covariance matrix $\Sigma$ of a $p$-dimensional normal random vector based on $n$ independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of the noise without any sparsity constraint on its covariance matrix. In this high-dimensional semiparametric deconvolution problem, we propose spectral thresholding estimators that are adaptive to the sparsity of $\Sigma$. We establish an oracle inequality for these estimators under model miss-specification and derive non-asymptotic minimax convergence rates that are shown to be logarithmic in $n/\log p$. We also discuss the estimation of low-rank matrices based on indirect observations as well as the generalization to elliptical distributions. The finite sample performance of the threshold estimators is illustrated in a numerical example.