Modern datasets are trending towards ever higher dimension. In response, recent theoretical studies of covariance estimation often assume the proportional-growth asymptotic framework, where the sample size $n$ and dimension $p$ are comparable, with $n, p \rightarrow \infty $ and $\gamma_n = p/n \rightarrow \gamma > 0$. Yet, many datasets -- perhaps most -- have very different numbers of rows and columns. We consider instead the disproportional-growth asymptotic framework, where $n, p \rightarrow \infty$ and $\gamma_n \rightarrow 0$ or $\gamma_n \rightarrow \infty$. Either disproportional limit induces novel behavior unseen within previous proportional and fixed-$p$ analyses. We study the spiked covariance model, with theoretical covariance a low-rank perturbation of the identity. For each of 15 different loss functions, we exhibit in closed form new optimal shrinkage and thresholding rules. Our optimal procedures demand extensive eigenvalue shrinkage and offer substantial performance benefits over the standard empirical covariance estimator. Practitioners may ask whether to view their data as arising within (and apply the procedures of) the proportional or disproportional frameworks. Conveniently, it is possible to remain {\it framework agnostic}: one unified set of closed-form shrinkage rules (depending only on the aspect ratio $\gamma_n$ of the given data) offers full asymptotic optimality under either framework. At the heart of the phenomena we explore is the spiked Wigner model, in which a low-rank matrix is perturbed by symmetric noise. Exploiting a connection to the spiked covariance model as $\gamma_n \rightarrow 0$, we derive optimal eigenvalue shrinkage rules for estimation of the low-rank component, of independent and fundamental interest.

翻译：现代数据集正趋向于更高维度。为此，近期协方差估计的理论研究通常采用比例增长渐近框架，其中样本量$n$与维度$p$相当，且$n, p \rightarrow \infty$，同时$\gamma_n = p/n \rightarrow \gamma > 0$。然而，许多数据集（或许大多数）的行列数量差异悬殊。本文转而考虑非比例增长渐近框架，其中$n, p \rightarrow \infty$且$\gamma_n \rightarrow 0$或$\gamma_n \rightarrow \infty$。这两种非比例极限均会引发先前比例分析与固定$p$分析中未出现的新行为。我们研究尖峰协方差模型，其理论协方差为单位矩阵的低秩扰动。针对15种不同的损失函数，我们以闭合形式给出了新的最优收缩与阈值规则。我们的最优方法要求大幅度的特征值收缩，并在性能上显著优于标准经验协方差估计量。实践者可能会质疑是否应将数据视为源自比例框架（并应用相应方法）还是非比例框架。便利的是，可以保持"框架无关性"：一组统一的闭合形式收缩规则（仅依赖于给定数据的长宽比$\gamma_n$）在两种框架下均具有完全渐近最优性。我们探究的现象核心是尖峰Wigner模型，其中低秩矩阵被对称噪声扰动。利用当$\gamma_n \rightarrow 0$时与尖峰协方差模型的关联，我们推导出用于估计低秩成分的最优特征值收缩规则，这具有独立且基础性的重要意义。