Statistics Papers

Document Type

Journal Article

Date of this Version

2012

Publication Source

The Annals of Statistics

Volume

40

Issue

5

Start Page

2389

Last Page

2420

DOI

10.1214/12-AOS998

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 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.

Keywords

Assouad’s lemma, Bregman divergence, covariance matrix estimation, Frobenius norm, Le Cam’s method, minimax lower bound, spectral norm, optimal rate of convergence, thresholding

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Date Posted: 27 November 2017

This document has been peer reviewed.