Optimal Kernel Learning for Gaussian Process Models with High-Dimensional Input

TL;DR

This paper introduces an optimal kernel learning method for Gaussian process regression that effectively identifies active variables, reduces computational costs, and improves prediction accuracy in high-dimensional input scenarios.

Contribution

It proposes a novel convex combination approach for covariance functions, inspired by optimal design principles, incorporating the effect heredity principle for sparse active variable selection.

Findings

Outperforms existing methods in identifying active variables

Enhances prediction accuracy in high-dimensional settings

Simplifies complex system interpretation

Abstract

Gaussian process (GP) regression is a popular surrogate modeling tool for computer simulations in engineering and scientific domains. However, it often struggles with high computational costs and low prediction accuracy when the simulation involves too many input variables. For some simulation models, the outputs may only be significantly influenced by a small subset of the input variables, referred to as the ``active variables''. We propose an optimal kernel learning approach to identify these active variables, thereby overcoming GP model limitations and enhancing system understanding. Our method approximates the original GP model's covariance function through a convex combination of kernel functions, each utilizing low-dimensional subsets of input variables. Inspired by the Fedorov-Wynn algorithm from optimal design literature, we develop an optimal kernel learning algorithm to determine this approximation. We incorporate the effect heredity principle, a concept borrowed from the field of ``design and analysis of experiments'', to ensure sparsity in active variable selection. Through several examples, we demonstrate that the proposed method outperforms alternative approaches in correctly identifying active input variables and improving prediction accuracy. It is an effective solution for interpreting the surrogate GP regression and simplifying the complex underlying system.