Bayesian Quadrature, Energy Minimization, and Space-Filling Design

A standard objective in computer experiments is to approximate the behavior of an unknown function on a compact domain from a few evaluations inside the domain. When little is known about the function, space-filling design is advisable: typically, points of evaluation spread out across the available space are obtained by minimizing a geometrical (for instance, covering radius) or a discrepancy criterion measuring distance to uniformity. The paper investigates connections between design for integration (quadrature design), construction of the (continuous) BLUE for the location model, space-filling design, and minimization of energy (kernel discrepancy) for signed measures. Integrally strictly positive definite kernels define strictly convex energy functionals, with an equivalence between the notions of potential and directional derivative, showing the strong relation between discrepancy minimization and more traditional design of optimal experiments. In particular, kernel herding algorithms, which are special instances of vertex-direction methods used in optimal design, can be applied to the construction of point sequences with suitable space-filling properties.