OPTIMAL RATES OF CONVERGENCE FOR SPARSE COVARIANCE MATRIX ESTIMATION
OPTIMAL RATES OF CONVERGENCE FOR SPARSE COVARIANCE MATRIX ESTIMATION
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DOI：
10.1214/12-aos998
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发表时间：
2012-10-01
影响因子：
4.5
通讯作者：
Zhou, Harrison H.
中科院分区：
文献类型：
--
作者：
Cai, T. Tony;Zhou, Harrison H.
本文考虑了稀疏协方差矩阵的估计，建立了在一定范围的矩阵算子范数和Bregman散度损失下的最优收敛速率。一个主要的焦点是对速率急剧极小极大下界的推导。该问题呈现出与传统非参数函数估计问题明显不同的新特征。标准技术不能产生好的结果，因此需要新的工具。我们首先开发了一种下界技术，特别适合于处理“双向”问题，如估计稀疏协方差矩阵。结果可以看作是勒卡姆方法在一个方向上的推广和阿苏德引理在另一个方向上的推广。这种下界技术具有独立的意义，可用于其他矩阵估计问题。然后应用一般下界技术建立了谱范数下估计稀疏协方差矩阵的速率锐极小下界。给出了一种阈值估计方法，在谱范数下得到了最优的收敛速率。然后将结果推广到1的一般矩阵l(w)算子范数
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 l(w) operator norms for 1