Gaussian process regression with log-linear scaling for common non-stationary kernels

2024/07/04 by Kielstra, P. Michael, Lindsey, Michael
#Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2407.03608

Abstract

We introduce a fast algorithm for Gaussian process regression in low dimensions, applicable to a widely-used family of non-stationary kernels. The non-stationarity of these kernels is induced by arbitrary spatially-varying vertical and horizontal scales. In particular, any stationary kernel can be accommodated as a special case, and we focus especially on the generalization of the standard Matérn kernel. Our subroutine for kernel matrix-vector multiplications scales almost optimally as O(Nlog N), where N is the number of regression points. Like the recently developed equispaced Fourier Gaussian process (EFGP) methodology, which is applicable only to stationary kernels, our approach exploits non-uniform fast Fourier transforms (NUFFTs). We offer a complete analysis controlling the approximation error of our method, and we validate the method's practical performance with numerical experiments. In particular we demonstrate improved scalability compared to to state-of-the-art rank-structured approaches in spatial dimension d>1.

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