Efficient Gaussian Process Calculations Using Chebyshev Nodes and Fast Fourier Transform

Abstract

Gaussian processes have gained popularity in contemporary solutions for mathematical modeling problems, particularly in cases involving complex and challenging-to-model scenarios or instances with a general lack of data. Therefore, they often serve as generative models for data, for example, in classification problems. However, a common problem in the application of Gaussian processes is their computational complexity. To address this challenge, sparse methods are frequently employed, involving a reduction in the computational domain. In this study, we propose an innovative computational approach for Gaussian processes. Our method revolves around selecting a computation domain based on Chebyshev nodes, with the optimal number of nodes determined by minimizing the degree of the Chebyshev series, while ensuring meaningful coefficients derived from function values at the Chebyshev nodes with fast Fourier transform. This approach not only facilitates a reduction in computation time but also provides a means to reconstruct the original function using the functional series. We conducted experiments using two computational methods for Gaussian processes: Markov chain Monte Carlo and integrated nested Laplace approximation. The results demonstrate a significant reduction in computation time, thereby motivating further development of the proposed algorithm.