We introduce a Fourier-based fast algorithm for Gaussian process regression in low dimensions. It approximates a translationally-invariant covariance kernel by complex exponentials on an equispaced Cartesian frequency grid of $M$ nodes. This results in a weight-space $M\times M$ system matrix with Toeplitz structure, which can thus be applied to a vector in ${\mathcal O}(M \log{M})$ operations via the fast Fourier transform (FFT), independent of the number of data points $N$. The linear system can be set up in ${\mathcal O}(N + M \log{M})$ operations using nonuniform FFTs. This enables efficient massive-scale regression via an iterative solver, even for kernels with fat-tailed spectral densities (large $M$). We provide bounds on both kernel approximation and posterior mean errors. Numerical experiments for squared-exponential and Mat\'ern kernels in one, two and three dimensions often show 1-2 orders of magnitude acceleration over state-of-the-art rank-structured solvers at comparable accuracy. Our method allows 2D Mat\'ern-$\mbox{$\frac{3}{2}$}$ regression from $N=10^9$ data points to be performed in 2 minutes on a standard desktop, with posterior mean accuracy $10^{-3}$. This opens up spatial statistics applications 100 times larger than previously possible.

翻译：我们提出一种基于傅里叶变换的低维高斯过程回归快速算法。该方法通过等距笛卡尔频率网格上$M$个节点的复指数近似平移不变协方差核，得到具有Toeplitz结构的权空间$M\times M$系统矩阵。利用快速傅里叶变换(FFT)，该矩阵可在${\mathcal O}(M \log M)$运算量内作用于向量，且与数据点数量$N$无关。通过非均匀FFT，线性系统可在${\mathcal O}(N + M \log M)$运算量内构建。这使得即使对于厚尾谱密度（大$M$）的核函数，也能通过迭代求解器实现高效大规模回归。我们给出了核近似误差和后验均值误差的界。在一维、二维和三维空间中针对平方指数核与Matérn核的数值实验表明，在同等精度下，本方法比最先进的秩结构求解器通常快1-2个数量级。在标准台式机上，我们的方法可在2分钟内完成基于$N=10^9$个数据点的二维Matérn-$\mbox{$\frac{3}{2}$}$回归，后验均值精度达$10^{-3}$。这使得空间统计应用规模比先前可行方案扩大100倍。