Extension of Gaussian-kernel variance bounds to generic point configurations

Establish whether the conditional-variance estimates for Gaussian-process interpolation with the Gaussian kernel in dimensions greater than one remain valid in some form for almost any sequence of input-point configurations, rather than only for configurations containing a Cartesian product grid.

Background

The paper proves factorial-rate upper bounds for the conditional variance of Gaussian-process interpolation with the Gaussian kernel when the input set contains a d-fold Cartesian product of one-dimensional point sets. The proof relies on tensor-product arguments and therefore applies directly to grid-like configurations.

The authors suggest that analogous estimates should hold for “almost any” input configuration in dimensions greater than one, but they do not establish this generalization. Such a result would clarify whether the severe global overconfidence identified for grid configurations is typical rather than dependent on specially structured sampling designs.