We introduce a fast algorithm for Gaussian process regression in low dimensions, applicable to a widely-used family of non-stationary kernels. The non-stationarity of these kernels is induced by arbitrary spatially-varying vertical and horizontal scales. In particular, any stationary kernel can be accommodated as a special case, and we focus especially on the generalization of the standard Mat\'ern kernel. Our subroutine for kernel matrix-vector multiplications scales almost optimally as $O(N\log N)$, where $N$ is the number of regression points. Like the recently developed equispaced Fourier Gaussian process (EFGP) methodology, which is applicable only to stationary kernels, our approach exploits non-uniform fast Fourier transforms (NUFFTs). We offer a complete analysis controlling the approximation error of our method, and we validate the method's practical performance with numerical experiments. In particular we demonstrate improved scalability compared to to state-of-the-art rank-structured approaches in spatial dimension $d>1$.

翻译：本文提出了一种适用于低维高斯过程回归的快速算法，该方法可广泛应用于一类常见的非平稳核函数。这些核函数的非平稳性源于任意空间变化的垂直与水平尺度参数。特别地，任何平稳核均可作为特例被纳入本框架，我们尤其关注标准Matérn核的广义化形式。我们实现的核矩阵-向量乘法子程序具有近乎最优的计算复杂度$O(N\log N)$，其中$N$为回归点数量。与近期仅适用于平稳核的等间距傅里叶高斯过程（EFGP）方法类似，本方法利用了非均匀快速傅里叶变换（NUFFT）技术。我们提供了完整的理论分析以控制方法的近似误差，并通过数值实验验证了其实际性能。特别地，在空间维度$d>1$的情况下，本方法相较于当前最先进的秩结构化方法展现出更优的可扩展性。