We show that in a common high-dimensional covariance model, the choice ofloss function has a profound effect on optimal estimation. In an asymptoticframework based on the Spiked Covariance model and use of orthogonallyinvariant estimators, we show that optimal estimation of the populationcovariance matrix boils down to design of an optimal shrinker $\eta$ that actselementwise on the sample eigenvalues. Indeed, to each loss function therecorresponds a unique admissible eigenvalue shrinker $\eta^*$ dominating allother shrinkers. The shape of the optimal shrinker is determined by the choiceof loss function and, crucially, by inconsistency of both eigenvalues andeigenvectors of the sample covariance matrix. Details of these phenomena andclosed form formulas for the optimal eigenvalue shrinkers are worked out for amenagerie of 26 loss functions for covariance estimation found in theliterature, including the Stein, Entropy, Divergence, Frechet,Bhattacharya/Matusita, Frobenius Norm, Operator Norm, Nuclear Norm andCondition Number losses.
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机译：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优估计有深远的影响。在基于尖峰协方差模型和使用正交不变估计器的渐近框架中，我们表明总体协方差矩阵的最佳估计归结为设计了一个最优收缩器$ \ eta $，该收缩器对样本特征值起作用。实际上，对于每个损失函数，对应于一个独占的可支配的特征值收缩器$ \ eta ^ * $支配所有其他收缩器。最佳收缩器的形状取决于损失函数的选择，最重要的是取决于样本协方差矩阵的特征值和特征向量的不一致。针对文学中发现的26种损失函数进行协方差估计，得出了这些现象的详细信息和最佳特征值收缩器的封闭形式公式，包括斯坦因，熵，发散，弗雷谢特，巴塔查亚/马托西塔，弗罗宾纽斯范数，算子范数，核范数和条件编号损失。
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