论文标题
提高了操作员规范的自举近似速率：无坐标方法
Improved Rates of Bootstrap Approximation for the Operator Norm: A Coordinate-Free Approach
论文作者
论文摘要
令$ \hatς= \ frac {1} {n} \ sum_ {i = 1}^n x_i \ otimes x_i $表示样品协方差i.i.d. i.i.d.〜观察$ x_1，\ dots，\ dots，\ dots，d dots，x_n $，x_n $，在真实的希尔伯特空间中，并让$} \ \ \ \ \ \ \ \ \ \ \ c（x） x_1）$。本文的重点是了解引导程序可以如何近似运算符规范错误的分布$ \ sqrt n \ | \ | \hatς-= \ | _ \ | _ {\ text {op}} $，在设置中，在$σ$衰减为$σ$ decay的eigenValue as $σ$ decay为$λ_j（f） $β> 1/2 $。我们的主要结果表明，Bootstrap可以以$ n^{ - \ frac {β-1/2} {2β+4+4+ε}} $的分配速率以$ nifit of kolmogorov metric，以$ s的$ sirirov $ sirric，$ yrirov $ srric，bootstrap可以近似$ \ sqrt n \ | \ | \ | \ | \ | \ | \ | _ {\ text {op}} $，以$ n^{ - \ frac {β-1/2} {2β+4+4+ε}} $ sifit the Kolmogorov $ sirric，特别是，这表明Bootstrap可以在大型$β$的状态下实现接近$ n^{ - 1/2} $费率 - 在同一制度中，这在以前的接近$ n^{ - 1/6} $速率上大大改善。除了获得更快的速度外，我们的分析还基于无坐标技术利用了根本不同的观点。此外，我们的结果具有更大的一般性，我们提出了一个与高维欧几里得空间中椭圆形和Marčenko-Pastur模型兼容的模型，这可能具有独立的兴趣。
Let $\hatΣ=\frac{1}{n}\sum_{i=1}^n X_i\otimes X_i$ denote the sample covariance operator of centered i.i.d.~observations $X_1,\dots,X_n$ in a real separable Hilbert space, and let $Σ=\mathbb{E}(X_1\otimes X_1)$. The focus of this paper is to understand how well the bootstrap can approximate the distribution of the operator norm error $\sqrt n\|\hatΣ-Σ\|_{\text{op}}$, in settings where the eigenvalues of $Σ$ decay as $λ_j(Σ)\asymp j^{-2β}$ for some fixed parameter $β>1/2$. Our main result shows that the bootstrap can approximate the distribution of $\sqrt n\|\hatΣ-Σ\|_{\text{op}}$ at a rate of order $n^{-\frac{β-1/2}{2β+4+ε}}$ with respect to the Kolmogorov metric, for any fixed $ε>0$. In particular, this shows that the bootstrap can achieve near $n^{-1/2}$ rates in the regime of large $β$ -- which substantially improves on previous near $n^{-1/6}$ rates in the same regime. In addition to obtaining faster rates, our analysis leverages a fundamentally different perspective based on coordinate-free techniques. Moreover, our result holds in greater generality, and we propose a model that is compatible with both elliptical and Marčenko-Pastur models in high-dimensional Euclidean spaces, which may be of independent interest.
