ESTIMATING SPARSE PRECISION MATRIX: OPTIMAL RATES OF CONVERGENCE AND ADAPTIVE ESTIMATION
ESTIMATING SPARSE PRECISION MATRIX: OPTIMAL RATES OF CONVERGENCE AND ADAPTIVE ESTIMATION
复制标题
估计稀疏精度矩阵：最佳收敛率和自适应估计
DOI：
10.1214/13-aos1171
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发表时间：
2016-04-01
影响因子：
4.5
通讯作者：
Zhou, Harrison H.
中科院分区：
文献类型：
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作者：
Cai, T. Tony;Liu, Weidong;Zhou, Harrison H.
精度矩阵在多元分析的广泛应用中具有重要意义。本文考虑高维设置中稀疏精度矩阵的自适应极小极大估计。针对一系列矩阵范数损失建立了最佳收敛率。提出了一种基于自适应约束 l1 最小化的完全数据驱动估计器，并在参数空间集合上获得了其收敛速度。该估计器称为 ACLIME，易于实现并且在数值上表现良好。建立极小极大收敛速率的一个主要步骤是导出速率锐利下界。应用“双向”下界技术来获得极小极大下界。上限和下限共同产生稀疏精度矩阵估计的最佳收敛速率，并表明 ACLIME 估计器对于参数空间集合和一系列矩阵范数损失同时具有自适应极小最大速率最优。
Precision matrix is of significant importance in a wide range of applications in multivariate analysis. This paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting. Optimal rates of convergence are established for a range of matrix norm losses. A fully data driven estimator based on adaptive constrained l1 minimization is proposed and its rate of convergence is obtained over a collection of parameter spaces. The estimator, called ACLIME, is easy to implement and performs well numerically. A major step in establishing the minimax rate of convergence is the derivation of a rate-sharp lower bound. A “two-directional” lower bound technique is applied to obtain the minimax lower bound. The upper and lower bounds together yield the optimal rates of convergence for sparse precision matrix estimation and show that the ACLIME estimator is adaptively minimax rate optimal for a collection of parameter spaces and a range of matrix norm losses simultaneously.