Fast Adaptive Fourier Integration for Spectral Densities of Gaussian Processes

TL;DR

This paper introduces a fast, adaptive Fourier integration method for efficiently computing Gaussian process covariance functions from spectral densities, enabling scalable modeling of complex processes like wind velocities.

Contribution

The authors develop an adaptive integration framework combining high-order quadrature, nonuniform FFT, and error bounds for accurate, fast covariance evaluation from arbitrary spectral densities.

Findings

Achieves several orders of magnitude speedup over naive methods.

Enables evaluation of covariance functions at millions of points in seconds.

Facilitates maximum likelihood estimation for complex spectral models.

Abstract

The specification of a covariance function is of paramount importance when employing Gaussian process models, but the requirement of positive definiteness severely limits those used in practice. Designing flexible stationary covariance functions is, however, straightforward in the spectral domain, where one needs only to supply a positive and symmetric spectral density. In this work, we introduce an adaptive integration framework for efficiently and accurately evaluating covariance functions and their derivatives at irregular locations directly from \textit{any} continuous, integrable spectral density. In order to make this approach computationally tractable, we employ high-order panel quadrature, the nonuniform fast Fourier transform, and a Nyquist-informed panel selection heuristic, and derive novel algebraic truncation error bounds which are used to monitor convergence. As a result, we demonstrate several orders of magnitude speedup compared to naive uniform quadrature approaches, allowing us to evaluate covariance functions from slowly decaying, singular spectral densities at millions of locations to a user-specified tolerance in seconds on a laptop. We then apply our methodology to perform gradient-based maximum likelihood estimation using a previously numerically infeasible long-memory spectral model for wind velocities below the atmospheric boundary layer.