Regular Fourier Features for Nonstationary Gaussian Processes
This work provides a more efficient and robust method for simulating nonstationary Gaussian processes, which is beneficial for researchers and practitioners working with complex spatial-temporal data where stationarity assumptions do not hold.
This paper tackles the challenge of simulating nonstationary Gaussian processes, which traditionally scales cubically with sample locations. The authors propose regular Fourier features, a method that discretizes the spectral representation directly, preserving correlation structure without requiring probability assumptions, and yielding an efficient low-rank approximation.
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.