Optimal Rates of Convergence for Sparse Covariance Matrix Estimation

Thumbnail Image
Penn collection
Statistics Papers
Degree type
Assouad’s lemma
Bregman divergence
covariance matrix estimation
Frobenius norm
Le Cam’s method
minimax lower bound
spectral norm
optimal rate of convergence
Physical Sciences and Mathematics
Statistics and Probability
Grant number
Copyright date
Related resources
Cai, T. Tony
Zhou, Harrison H

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 ℓw operator norms for 1 ≤ w ≤ ∞. 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.

Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Publication date
Journal title
The Annals of Statistics
Volume number
Issue number
Publisher DOI
Journal Issue
Recommended citation