In this paper, we present a comprehensive analysis of the posterior covariance field in Gaussian processes, with applications to the posterior covariance matrix. The analysis is based on the Gaussian prior covariance but the approach also applies to other covariance kernels. Our geometric analysis reveals how the Gaussian kernel's bandwidth parameter and the spatial distribution of the observations influence the posterior covariance as well as the corresponding covariance matrix, enabling straightforward identification of areas with high or low covariance in magnitude. Drawing inspiration from the a posteriori error estimation techniques in adaptive finite element methods, we also propose several estimators to efficiently measure the absolute posterior covariance field, which can be used for efficient covariance matrix approximation and preconditioning. We conduct a wide range of experiments to illustrate our theoretical findings and their practical applications.

翻译：本文对高斯过程中的后验协方差场进行了全面分析，并将其应用于后验协方差矩阵。该分析基于高斯先验协方差，但所提出的方法同样适用于其他协方差核函数。我们的几何分析揭示了高斯核的带宽参数与观测点的空间分布如何影响后验协方差及其对应的协方差矩阵，从而能够直接识别协方差幅值较高或较低的区域。受自适应有限元方法中后验误差估计技术的启发，我们还提出了若干估计量以高效度量绝对后验协方差场，这些估计量可用于高效的协方差矩阵近似与预条件处理。我们通过大量实验验证了理论发现及其实际应用价值。