Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model.
Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model.
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DOI：
10.1214/17-aos1601
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发表时间：
2018-08
影响因子：
4.5
通讯作者：
Johnstone IM
中科院分区：
文献类型：
--
作者：
Donoho DL;Gavish M;Johnstone IM
我们表明，在常见的高维协方差模型中，损失函数的选择对最佳估计具有深远的影响。
在基于峰值协方差模型的不对称框架和正交估计量的使用中，我们表明对种群协方差矩阵的最佳估计归结为最佳收缩器η的设计，这些估计确实在样品特征功能上起作用。那里对应于独特的特征值收缩η*主导所有其他收缩器。函数，完全通过样品协方差矩阵的特征值和特征向量的不一致。
这些现象的详细信息和最佳特征值收缩器的封闭式公式的详细信息是为26个损失功能的Menagerie制定的，用于文献中发现的协方差估计，包括Stein，Stein，Entropy，Divergence，Fréchet，Fréchet，Bhattacharya/bhattacharya/Matusita，Frobenius Norm，Frobenius Norm，Operator Norm Norm Norm， ，核标准和条件数损失。
We show that in a common high-dimensional covariance model, the choice of loss function has a profound effect on optimal estimation. In an asymptotic framework based on the Spiked Covariance model and use of orthogonally invariant estimators, we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker η that acts elementwise on the sample eigenvalues. Indeed, to each loss function there corresponds a unique admissible eigenvalue shrinker η* dominating all other shrinkers. The shape of the optimal shrinker is determined by the choice of loss function and, crucially, by inconsistency of both eigenvalues and eigenvectors of the sample covariance matrix. Details of these phenomena and closed form formulas for the optimal eigenvalue shrinkers are worked out for a menagerie of 26 loss functions for covariance estimation found in the literature, including the Stein, Entropy, Divergence, Fréchet, Bhattacharya/Matusita, Frobenius Norm, Operator Norm, Nuclear Norm and Condition Number losses.