Gaussian random field approximation via Stein's method with applications to wide random neural networks

Understand

We derive upper bounds on the Wasserstein distance ($W_1$), with respect to $\sup$-norm, between any continuous $\mathbb{R}^d$ valued random field indexed by the $n$-sphere and the Gaussian, based on Stein's method.

• We develop a novel Gaussian smoothing technique that allows us to transfer a bound in a smoother metric to the $W_1$ distance.
• The smoothing is based on covariance functions constructed using powers of Laplacian operators, designed so that the associated Gaussian process has a tractable Cameron-Martin or Reproducing Kernel Hilbert Space.
• This feature enables us to move beyond one dimensional interval-based index sets that were previously considered in the literature.