Lengthscale-informed sparse grids for high dimensional Gaussian process emulation

Abstract

Kernel interpolation, particularly in the context of Gaussian process emulation, is a powerful tool for surrogate modelling, allowing complex input–output maps to be approximated from relatively few function evaluations. However, in high-dimensional settings, these methods rapidly succumb to the curse of dimensionality---where the number of required function evaluations typically grows exponentially with respect to the input dimension. Fortunately, in many practical applications, functions often exhibit anisotropy in their parameters; where the influence of input parameters on the output can vary greatly. Different formulations of such functional anisotropy are routinely exploited in the high-dimensional approximation literature to mitigate dimensional dependence in the error, enabling the practical approximation of very high dimensional functions.

In this talk, we introduce a generalisation of sparse grid methods for kernel interpolation that incorporates anisotropy through the lengthscale parameter of Matérn kernels, inspired by the empirical success of hyperparameter estimation in high-dimensional Gaussian process regression. We present error bounds demonstrating reduced dependence on the input dimension, together with numerical experiments demonstrating accurate interpolation in very high dimensions when sufficient anisotropy is present.