Optimal rates of convergence for sparse covariance matrix estimation

TL;DR

This paper establishes the optimal convergence rates for estimating sparse covariance matrices across various norms, introducing a novel lower bound technique and demonstrating the effectiveness of thresholding estimators.

Contribution

It develops a new lower bound technique for matrix estimation problems and determines the optimal rates of convergence for sparse covariance matrices under multiple loss functions.

Findings

Thresholding estimators achieve optimal convergence rates.

New lower bound technique applicable to matrix estimation.

Results extend to various matrix operator norms and Bregman divergences.

Abstract

This paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and Bregman divergence losses. A major focus is on the derivation of a rate sharp minimax lower bound. The problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems. Standard techniques fail to yield good results, and new tools are thus needed. We first develop a lower bound technique that is particularly well suited for treating "two-directional" problems such as estimating sparse covariance matrices. The result can be viewed as a generalization of Le Cam's method in one direction and Assouad's Lemma in another. This lower bound technique is of independent interest and can be used for other matrix estimation problems. We then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique. A thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm. The results are then extended to the general matrix $\ell_w$ operator norms for $1\le w\le \infty$. In addition, we give a unified result on the minimax rate of convergence for sparse covariance matrix estimation under a class of Bregman divergence losses.