2012/10/01 by T. Tony Cai, Harrison H. Zhou · 147 citations
Computer Science · Engineering · Mathematics · #Covariance #Covariance matrix #Direction-of-Arrival Estimation Techniques #Distributed Sensor Networks and Detection Algorithms #Estimation of covariance matrices #Estimator #Matrix norm #Minimax #Rate of convergence #Sparse and Compressive Sensing Techniques #Upper and lower bounds #math.ST #stat.TH
paper · pdf · doi:10.1214/12-aos998
published in The Annals of Statistics 40(5) (Institute of Mathematical Statistics) · Published in at http://dx.doi.org/10.1214/12-AOS998 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2012/10/01 · arxiv created 2013/02/13 · arxiv updated 2013/02/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.