Gaussian approximation for the sup-norm of high-dimensional matrix-variate U-statistics and its applications

We study the Gaussian approximation of high-dimensional U-statistics of order two under the supremum norm. For non-degenerate U-statistics, we propose a two-step Gaussian approximation procedure and establish the explicit rate of convergence that decays polynomially in sample size. We also supplement a practical Gaussian wild bootstrap method to approximate the quantiles of the maxima of centered U-statistics and prove its asymptotic validity under a high-dimensional scaling limit. Our theoretical results are demonstrated on several statistical applications involving adaptive tuning parameter selection, simultaneous inference and related functional estimation of the covariance matrix for high-dimensional non-Gaussian data.