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.

翻译：模拟高斯过程需要从高维高斯分布中采样，其计算复杂度随采样位置数量呈立方增长。谱方法通过利用傅里叶表示来应对这一挑战，将谱密度视为蒙特卡洛近似的概率分布。尽管这种概率解释适用于平稳过程，但对于非平稳情形则限制过强——此时谱密度通常不再是概率测度。我们针对可调和过程提出了规则傅里叶特征以突破此限制。该方法直接对谱表示进行离散化处理，在无需概率假设的前提下保持谱权重间的相关结构。在有限谱支撑集的假设下，该方法可构造出高效的低秩近似，且其半正定性由构造方式自然保证。当谱密度未知时，该框架可自然地扩展至基于数据的核函数学习。我们在局部平稳核与具有复值谱密度的可调和混合核上验证了该方法的有效性。