Bootstrapping the Operator Norm in High Dimensions: Error Estimation for Covariance Matrices and Sketching

Abstract: Although the operator (spectral) norm is one of the most widely used metrics for covariance estimation, comparatively little is known about the fluctuations of error in this norm. To be specific, let $\hat\Sigma$ denote the sample covariance matrix of $n$ observations in $\mathbb{R}<sup>p$ that arise from a population matrix $\Sigma$, and let $T_n=\sqrt{n}|\hat\Sigma-\Sigma|_{\text{op}}$. In the setting where the eigenvalues of $\Sigma$ have a decay profile of the form $\lambda_j(\Sigma)\asymp j<sup>{-2\beta}$, we analyze how well the bootstrap can approximate the distribution of $T_n$. Our main result shows that up to factors of $\log(n)$, the bootstrap can approximate the distribution of $T_n$ at the dimension-free rate of $n<sup>{-\frac{\beta-1/2}{6\beta+4}}$, with respect to the Kolmogorov metric. Perhaps surprisingly, a result of this type appears to be new even in settings where $p&lt; n$. More generally, we discuss the consequences of this result beyond covariance matrices and show how the bootstrap can be used to estimate the errors of sketching algorithms in randomized numerical linear algebra (RandNLA). An illustration of these ideas is also provided with a climate data example.