论文标题
在半圆极限的最佳特征值收缩
Optimal Eigenvalue Shrinkage in the Semicircle Limit
论文作者
论文摘要
现代数据集朝着更高的维度发展。作为回应，最近对协方差估计的理论研究通常假定比例增长的渐近框架，其中样本量$ n $和dimension $ p $是可比性的，其中$ n，p \ rightarrow \ rightarrow \ infty $ and $γ_n= p/n \ p/n \rightarrowγ> 0 $。但是，许多数据集（也许大多数）具有非常不同的行和列数。我们考虑的是，$ n，p \ rightarrow \ infty $和$γ_n\ rightarrow 0 $或$γ_n\ rightarrow \ rightarrow \ infty $。在以前的比例和固定$ $ p $分析中，限制不成比例的限制会引起新的行为。我们研究了尖峰协方差模型，理论协方差是身份的低级别扰动。对于15个不同的损失功能，我们以封闭形式展示了新的最佳收缩和阈值规则。我们的最佳程序需要广泛的特征值收缩，并比标准的经验协方差估计器具有可观的性能优势。 从业者可以询问是否将其数据视为（并应用）比例或不正确框架的过程中产生的数据。方便地，可以保持{\ it框架不可知}：一组统一的封闭形式收缩规则（仅取决于给定数据的宽高比$γ_n$）在任何一个框架下都提供完整的渐近最佳性。我们探索的现象的核心是尖刺的Wigner模型，其中低级别基质受到对称噪声的干扰。利用与峰值协方差模型的连接为$γ_n\ rightarrow 0 $，我们得出了最佳的特征值收缩规则，以估计低级别组件的独立和基本利益。
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 $γ_n = p/n \rightarrow γ> 0$. 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 $γ_n \rightarrow 0$ or $γ_n \rightarrow \infty$. 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 $γ_n$ 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 $γ_n \rightarrow 0$, we derive optimal eigenvalue shrinkage rules for estimation of the low-rank component, of independent and fundamental interest.
