High-dimensional covariance matrix estimation with missing observations

TL;DR

This paper introduces a computationally efficient method for estimating high-dimensional covariance matrices with missing data, achieving near-optimal rates without data imputation, and provides theoretical guarantees and bounds.

Contribution

It proposes a novel, practical covariance estimation procedure for high-dimensional data with missing observations, with proven optimality bounds.

Findings

Establishes non-asymptotic oracle inequalities for estimation accuracy.

Proves minimax optimality of the proposed rates up to a logarithmic factor.

Provides a computationally feasible approach that does not require data imputation.

Abstract

In this paper, we study the problem of high-dimensional approximately low-rank covariance matrix estimation with missing observations. We propose a simple procedure computationally tractable in high-dimension and that does not require imputation of the missing data. We establish non-asymptotic sparsity oracle inequalities for the estimation of the covariance matrix with the Frobenius and spectral norms, valid for any setting of the sample size and the dimension of the observations. We further establish minimax lower bounds showing that our rates are minimax optimal up to a logarithmic factor.