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

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 hatSigma denote the sample covariance matrix of n observations in ℝp that arise from a population matrix Sigma, and let Tn=√n|hatSigma-Sigma|op. In the setting where the eigenvalues of Sigma have a decay profile of the form λj(Sigma)asymp j-2β, we analyze how well the bootstrap can approximate the distribution of Tn. Our main result shows that up to factors of log(n), the bootstrap can approximate the distribution of Tn at the dimension-free rate of n^-fracβ-1/26β+4, with respect to the Kolmogorov metric. Perhaps surprisingly, a result of this type appears to be new even in settings where p< 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.