🤖 AI Summary
Simulating high-dimensional nonstationary Gaussian processes is challenging due to high computational complexity, and conventional spectral methods are limited by the assumption that the spectral density must be a probability measure, which hinders their applicability to nonstationary settings. This work proposes a regularized Fourier features approach that directly discretizes the spectral representation of harmonizable processes, yielding complex-valued Fourier features without requiring the spectral density to be a probability measure. The method enables efficient low-rank approximations while preserving the correlation structure encoded in spectral weights. It naturally supports learning kernel functions from data and demonstrates strong empirical performance on both locally stationary and harmonizable mixture kernels, significantly improving the efficiency and structural fidelity of nonstationary Gaussian process simulation.
📝 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.