The high efficiency of a recently proposed method for computing with Gaussian processes relies on expanding a (translationally invariant) covariance kernel into complex exponentials, with frequencies lying on a Cartesian equispaced grid. Here we provide rigorous error bounds for this approximation for two popular kernels—Matérn and squared exponential—in terms of the grid spacing and size. The kernel error bounds are uniform over a hypercube centered at the origin. Our tools include a split into aliasing and truncation errors, and bounds on sums of Gaussians or modified Bessel functions over various lattices. For the Matérn case, motivated by numerical study, we conjecture a stronger Frobenius-norm bound on the covariance matrix error for randomly-distributed data points. Lastly, we prove bounds on, and study numerically, the ill-conditioning of the linear systems arising in such regression problems.

中文翻译：

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使用等距傅里叶网格对常见高斯过程核进行均匀逼近

最近提出的高斯过程计算方法的高效率依赖于将（平移不变）协方差核扩展为复指数，频率位于笛卡尔等距网格上。在这里，我们根据网格间距和大小为两种流行的内核（Matérn 和平方指数）的近似提供了严格的误差范围。核误差界在以原点为中心的超立方体上是均匀的。我们的工具包括对混叠误差和截断误差的划分，以及各种格上高斯总和或修正贝塞尔函数的界限。对于 Matérn 案例，受数值研究的启发，我们推测随机分布数据点的协方差矩阵误差有更强的 Frobenius 范数界。最后，我们证明了此类回归问题中出现的线性系统病态的界限并进行了数值研究。