论文标题
经验光谱投影仪的定量极限定理和自举近似
Quantitative limit theorems and bootstrap approximations for empirical spectral projectors
论文作者
论文摘要
给定有限的I.I.D.〜样品在Hilbert空间中具有零平均值和Trace-Class协方差运算符$σ$，因此在许多应用中恢复了$σ$的光谱投影仪的问题。在本文中，我们考虑了找到经验协方差算子$ \hatσ$的光谱投影仪的分布近似的问题，并提供了一个无维度的框架，其中复杂性的特征在于所谓的相对等级为$σ$。在这种情况下，新颖的定量极限定理和自举近似仅在矩和光谱衰减方面仅受温和条件。在许多情况下，这些甚至可以改善高斯环境中的现有结果。
Given finite i.i.d.~samples in a Hilbert space with zero mean and trace-class covariance operator $Σ$, the problem of recovering the spectral projectors of $Σ$ naturally arises in many applications. In this paper, we consider the problem of finding distributional approximations of the spectral projectors of the empirical covariance operator $\hat Σ$, and offer a dimension-free framework where the complexity is characterized by the so-called relative rank of $Σ$. In this setting, novel quantitative limit theorems and bootstrap approximations are presented subject only to mild conditions in terms of moments and spectral decay. In many cases, these even improve upon existing results in a Gaussian setting.
