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 -- Mat\'ern 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 Mat\'ern 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.

翻译：近期提出的一种用于高斯过程计算的高效方法，依赖于将（平移不变的）协方差核展开为复指数函数，其频率位于笛卡尔等距网格上。本文针对两种常用核函数——Matérn核与平方指数核——在网格间距与规模方面提供了该近似的严格误差界。核误差界在原点为中心的超立方体上具有均匀性。我们的工具包括将误差分解为混叠误差与截断误差两部分，以及各类格点上高斯函数或修正贝塞尔函数之和的界。针对Matérn情形，基于数值研究的启发，我们提出关于随机分布数据点协方差矩阵误差的更优Frobenius范数界的猜想。最后，我们分析了此类回归问题中线性系统病态性的界，并进行了数值研究。