English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Although the operator (spectral) norm is one of the most widely used metricsfor covariance estimation, comparatively little is known about the fluctuationsof error in this norm. To be specific, let $\hat\Sigma$ denote the samplecovariance matrix of $n$ observations in $\mathbb{R}^p$ that arise from apopulation matrix $\Sigma$, and let$T_n=\sqrt{n}\|\hat\Sigma-\Sigma\|_{\text{op}}$. In the setting where theeigenvalues of $\Sigma$ have a decay profile of the form$\lambda_j(\Sigma)\asymp j^{-2\beta}$, we analyze how well the bootstrap canapproximate the distribution of $T_n$. Our main result shows that up to factorsof $\log(n)$, the bootstrap can approximate the distribution of $T_n$ at thedimension-free rate of $n^{-\frac{\beta-1/2}{6\beta+4}}$, with respect to theKolmogorov metric. Perhaps surprisingly, a result of this type appears to benew even in settings where $p< n$. More generally, we discuss the consequencesof this result beyond covariance matrices and show how the bootstrap can beused to estimate the errors of sketching algorithms in randomized numericallinear algebra (RandNLA). An illustration of these ideas is also provided witha climate data example.

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