Regular Fourier Features for Nonstationary Gaussian Processes

TL;DR

This paper introduces regular Fourier features for nonstationary Gaussian processes, enabling efficient spectral approximation without probability constraints, and extends to kernel learning from data.

Contribution

It proposes a novel discretization of spectral representation for nonstationary processes, avoiding probability assumptions and allowing low-rank, positive semi-definite approximations.

Findings

Efficient low-rank approximation for nonstationary Gaussian processes.

Extension to kernel learning when spectral density is unknown.

Successful application to locally stationary and harmonizable mixture kernels.

Abstract

Simulating a Gaussian process requires sampling from a high-dimensional Gaussian distribution, which scales cubically with the number of sample locations. Spectral methods address this challenge by exploiting the Fourier representation, treating the spectral density as a probability distribution for Monte Carlo approximation. Although this probabilistic interpretation works for stationary processes, it is overly restrictive for the nonstationary case, where spectral densities are generally not probability measures. We propose regular Fourier features for harmonizable processes that avoid this limitation. Our method discretizes the spectral representation directly, preserving the correlation structure among spectral weights without requiring probability assumptions. Under a finite spectral support assumption, this yields an efficient low-rank approximation that is positive semi-definite by construction. When the spectral density is unknown, the framework extends naturally to kernel learning from data. We demonstrate the method on locally stationary kernels and on harmonizable mixture kernels with complex-valued spectral densities.