Rates of Bootstrap Approximation for Eigenvalues in High-Dimensional PCA

In the context of principal components analysis (PCA), bootstrap is commonly applied to solve a variety inference problems, such as constructing confidence intervals for eigenvalues population covariance matrix $\Sigma$. However, when data are high-dimensional, there relatively few theoretical guarantees that quantify performance bootstrap. Our aim in this paper analyze how well can approximate joint distribution leading sample $\hat\Sigma$, and we establish non-asymptotic rates approximation with respect multivariate Kolmogorov metric. Under certain assumptions, show achieve dimension-free rate ${\tt{r}}(\Sigma)/\sqrt n$ up logarithmic factors, where ${\tt{r}}(\Sigma)$ effective rank $\Sigma$, $n$ size. From methodological standpoint, our work also illustrates applying transformation $\hat\Sigma$ before bootstrapping an important consideration high-dimensional settings.