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.

翻译：令 $Z_1,\ldots,Z_n$ 为 $\mathbb{R}^p$ 中的独立同分布各向同性随机向量，$T \subset \mathbb{R}^p$ 为一紧集。经验过程理论的一个经典研究方向是通过一个称为 $T$ 的高斯宽度的单一参数来刻画二次过程上确界的大小：\begin{align*} \sup_{t \in T} \bigg| \frac{1}{n}\sum_{i=1}^n \langle Z_i,t \rangle^2-\lVert t \rVert^2 \bigg|, \end{align*}。本文针对标准高斯向量 $\{Z_i\}$ 引入了该二次过程上确界的一个改进界，该界对于 $T$ 的某些选择可以精确达到，因此被称为精确界。我们的精确界通过 $T$ 的球截面上一系列（随机）高斯宽度来表达，这构成了 $T$ 的高斯宽度的一种自然多尺度类比。与二次过程的经典界相比，我们的新界不仅确定了经典界中对某些 $T$ 可达到的最优常数，而且精确捕捉了经典界无法触及的二次过程的某些微妙相变行为。为说明我们结果的实用性，我们得到了随机投影的高斯 Dvoretzky-Milman 定理和协方差估计的 Koltchinskii-Lounici 定理的紧版本，两者均具有最优常数。此外，我们的界恢复了样本协方差矩阵最大特征值的著名 BBP 相变行为及其向样本协方差误差的推广。我们结果的证明利用了近期得到锐化的高斯比较不等式。我们证明方法的技术范围进一步通过获得双侧 Chevet 不等式的精确界得到展示。