Gaussian processes (GPs) are a ubiquitous tool for geostatistical modeling with high levels of flexibility and interpretability, and the ability to make predictions at unseen spatial locations through a process called Kriging. Estimation of Kriging weights relies on the inversion of the process' covariance matrix, creating a computational bottleneck for large spatial datasets. In this paper, we propose an Amortized Bayesian Local Interpolation NetworK (A-BLINK) for fast covariance parameter estimation, which uses two pre-trained deep neural networks to learn a mapping from spatial location coordinates and covariance function parameters to Kriging weights and the spatial variance, respectively. The fast prediction time of these networks allows us to bypass the matrix inversion step, creating large computational speedups over competing methods in both frequentist and Bayesian settings, and also provides full posterior inference and predictions using Markov chain Monte Carlo sampling methods. We show significant increases in computational efficiency over comparable scalable GP methodology in an extensive simulation study with lower parameter estimation error. The efficacy of our approach is also demonstrated using a temperature dataset of US climate normals for 1991--2020 based on over 7,000 weather stations.

翻译：高斯过程（GPs）是一种广泛应用于地统计学建模的工具，具有高度的灵活性和可解释性，并能够通过克里金法对未见空间位置进行预测。克里金权重的估计依赖于过程协方差矩阵的求逆，这为大规模空间数据集带来了计算瓶颈。本文提出一种用于快速协方差参数估计的摊销贝叶斯局部插值网络（A-BLINK），该网络使用两个预训练的深度神经网络，分别学习从空间位置坐标与协方差函数参数到克里金权重及空间方差的映射关系。这些网络的快速预测时间使我们能够绕过矩阵求逆步骤，在频率主义和贝叶斯框架下均较竞争方法实现显著的计算加速，同时支持通过马尔可夫链蒙特卡罗采样方法进行完整的后验推断与预测。在广泛的模拟研究中，我们展示了该方法相较于可扩展高斯过程同类方法在计算效率上的显著提升，且参数估计误差更低。我们还基于美国1991-2020年间7,000多个气象站的气候常态温度数据集，验证了该方法的有效性。