论文标题
应用应用程序的某些二次经验过程的确切界限
Exact bounds for some quadratic empirical processes with applications
论文作者
论文摘要
令$ z_1，\ ldots，z_n $ be i.i.d. $ \ mathbb {r}^p $中的各向同性随机向量，$ t \ subset \ mathbb {r}^p $是一个紧凑的集合。经验过程的经典线条理论表征了二次过程的近视的大小\ begin {align*} \ sup_ {t \ in t} \ bigg | \ frac {1} {n} \ sum_ {i = 1}^n \ langle z_i，t \ rangle^2- \ lvert t \ rvert t \ rvert^2 \ bigg |，\ end end end {align {align {align {align {align {align*}，被称为$ t $ $ t $的高斯宽度。 本文为标准高斯矢量$ \ {z_i \} $介绍了该二次过程的上等范围的改进，该过程可用于某些$ t $的某些选择，因此被称为确切的绑定。我们的确切界限是通过（随机）高斯宽度集的$ t $的（随机）高斯宽度的集合来表达的，$ t $是天然的多尺度类似物，类似于$ t $的高斯宽度。与二次过程的经典界限相比，我们的新界限不仅确定了经典界限中可以在某些$ t $中获得的最佳常数，而且还确定了超出经典范围的二次过程的某些微妙的相位过渡行为。 为了说明结果的实用性，我们获得了随机投影的高斯Dvoretzky-Milman定理的紧密版本，以及Koltchinskii-Lounici定理进行协方差估计，均具有最佳常数。此外，我们的边界恢复了样本协方差的顶级特征值及其对样品协方差误差的概括。 我们结果的证明最近削减了高斯比较不平等。我们的证明方法的技术范围进一步证明了双面雪ve不平等的确切结合。
Let $Z_1,\ldots,Z_n$ be i.i.d. isotropic random vectors in $\mathbb{R}^p$, and $T \subset \mathbb{R}^p$ be a compact set. A classical line of empirical process theory characterizes the size of the suprema of the quadratic process \begin{align*} \sup_{t \in T} \bigg| \frac{1}{n}\sum_{i=1}^n \langle Z_i,t \rangle^2-\lVert t \rVert^2 \bigg|, \end{align*} via a single parameter known as the Gaussian width of $T$. This paper introduces an improved bound for the suprema of this quadratic process for standard Gaussian vectors $\{Z_i\}$ that can be exactly attained for certain choices of $T$, and is thus referred to as an exact bound. Our exact bound is expressed via a collection of (stochastic) Gaussian widths over spherical sections of $T$ that serves as a natural multi-scale analogue to the Gaussian width of $T$. Compared to the classical bounds for the quadratic process, our new bounds not only determine the optimal constants in the classical bounds that can be attained for some $T$, but also precisely capture certain subtle phase transitional behavior of the quadratic process beyond the reach of the classical bounds. To illustrate the utility of our results, we obtain tight versions of the Gaussian Dvoretzky-Milman theorem for random projection, and the Koltchinskii-Lounici theorem for covariance estimation, both with optimal constants. Moreover, our bounds recover the celebrated BBP phase transitional behavior of the top eigenvalue of the sample covariance and its generalization to the sample covariance error. The proof of our results exploits recently sharpened Gaussian comparison inequalities. The technical scope of our method of proof is further demonstrated in obtaining an exact bound for a two-sided Chevet inequality.
