Show an illustration, what does it do intuitively? Example of how to choose the wild bootstrap process: for instance, the statistical learning theory reader might wonder whether it makes sense to use a correlated Rademacher or Gaussian process here.

A wild bootstrap method for nonparametric hypothesis tests based on kernel distribution embeddings is proposed. This bootstrap method is used to construct provably consistent tests that apply to random processes, for which the naive permutation-based bootstrap fails. It applies to a large group of kernel tests based on V-statistics, which are degenerate under the null hypothesis, and non-degenerate elsewhere. To illustrate this approach, we construct a two-sample test, an instantaneous independence test and a multiple lag independence test for time series. In experiments, the wild bootstrap gives strong performance on synthetic examples, on audio data, and in performance benchmarking for the Gibbs sampler.

Stein's methods (Stein, 1972) have been widely used in the machine learning and

This procedure provides a solution to the fundamental kernel selection problem as we can aggregate a large number of kernels with several bandwidths without incurring a significant loss of test power.

We describe and analyze a computionally efficient refitting procedure for computing high-probability upper bounds on the instance-wise mean-squared prediction error of penalized nonparametric estimates based on least-squares minimization. Requiring only a single dataset and black box access to the prediction method, it consists of three steps: computing suitable residuals, symmetrizing and scaling them with a pre-factor $ρ$, and using them to define and solve a modified prediction problem recentered at the current estimate. We refer to it as wild refitting, since it uses Rademacher residual symmetrization as in a wild bootstrap variant. Under relatively mild conditions allowing for noise heterogeneity, we establish a high probability guarantee on its performance, showing that the wild refit with a suitably chosen wild noise scale $ρ$ gives an upper bound on prediction error. This theoretical analysis provides guidance into the design of such procedures, including how the residuals should be formed, the amount of noise rescaling in the wild sub-problem needed for upper bounds, and the local stability properties of the block-box procedure. We illustrate the applicability of this procedure to various problems, including non-rigid structure-from-motion recovery with structured matrix penalties; plug-and-play image restoration with deep neural network priors; and randomized sketching with kernel methods.