Kolmogorov-Arnold-Kahane emulation for computer experiments in high dimensions

Kolmogorov and Arnold’s superposition theorems are viewed as possible tools to break the curse of dimensionality for approximating high-dimensional functions. Based on Kahane’s superposition theorem, this paper proposes a modeling approach, called Kolmogorov-Arnold-Kahane (KAK) emulation, for high-dimensional computer experiments. We use splines to parameterize the one-dimensional functions in KAK emulation, and then present a block coordinate descent algorithm to estimate the parameters. The parameterization procedure results in a new sample size determination rule in computer experiments, which supports the widely-used 10d rule to some extent. Numerical experiments indicate that KAK emulation is generally better than Kriging for high dimensions in terms of prediction accuracy.