Exact spectral norm error of sample covariance

Let $X_1,\ldots,X_n$ be i.i.d. centered Gaussian vectors in $\mathbb{R}^p$ with covariance $\Sigma$, and let $\hat{\Sigma}\equiv n^{-1}\sum_{i=1}^n X_iX_i^\top$ be the sample covariance. A central object of interest in the non-asymptotic theory of sample covariance is the spectral norm error $||\hat{\Sigma}-\Sigma||$ of the sample covariance $\hat{\Sigma}$. In the path-breaking work of Koltchinskii and Lounici [KL17a], the `zeroth-order' magnitude of $||\hat{\Sigma}-\Sigma||$ is characterized by the dimension-free two-sided estimate $\mathbb{E} \{||\hat{\Sigma}-\Sigma||/||\Sigma||\}\asymp \sqrt{r(\Sigma)/n}+r(\Sigma)/n$, using the so-called effective rank $r(\Sigma)\equiv \mathrm{tr}(\Sigma)/||\Sigma||$. The goal of this paper is to provide a dimension-free first-order characterization for $||\hat{\Sigma}-\Sigma||$. We show that \begin{equation*} \bigg|\frac{\mathbb{E} \{||\hat{\Sigma}-\Sigma||/||\Sigma||\} }{\mathbb{E}\sup_{\alpha \in [0,1]}\{(\alpha+n^{-1/2}\mathscr{G}_{\Sigma}(h;\alpha))^2-\alpha^2\}}- 1\bigg| \leq \frac{C}{\sqrt{r(\Sigma)} }, \end{equation*} where $\{\mathscr{G}_{\Sigma}(h;\alpha): \alpha \in [0,1]\}$ are (stochastic) Gaussian widths over spherical slices of the (standardized) $\Sigma$-ellipsoid, playing the role of a first-order analogue to the zeroth-order characteristic $r(\Sigma)$. As an immediate application of the first-order characterization, we obtain a version of the Koltchinskii-Lounici bound with optimal constants. In the more special context of spiked covariance models, our first-order characterization reveals a new phase transition of $||\hat{\Sigma}-\Sigma||$ that exhibits qualitatively different behavior compared to the BBP phase transitional behavior of $||\hat{\Sigma}||$. A similar phase transition is also proved for the associated eigenvector.