Exact bounds for some quadratic empirical processes with applications

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.