矩阵场的SPDE方法：图表示
The SPDE Approach to Matérn Fields: Graph Representations
摘要：
本文利用随机偏微分方程的图形表示研究了高斯马尔可夫随机场对非平稳高斯场的逼近。我们建立的逼近误差保证的理论基础上的谱收敛的图拉普拉斯算子。建议的图形表示提供了一个推广的Matérn模型的非结构化点云，并促进推理和采样使用线性代数方法稀疏矩阵。此外，他们桥梁和统一贝叶斯逆问题，空间统计学和基于图形的机器学习的几个模型。我们通过这三个学科的例子证明，图形表示所揭示的统一性促进了它们之间的思想交流。
英文摘要：
This paper investigates Gaussian Markov random field approximations to nonstationary Gaussian fields using graph representations of stochastic partial differential equations. We establish approximation error guarantees building on the theory of spectral convergence of graph Laplacians. The proposed graph representations provide a generalization of the Matérn model to unstructured point clouds, and facilitate inference and sampling using linear algebra methods for sparse matrices. In addition, they bridge and unify several models in Bayesian inverse problems, spatial statistics and graph-based machine learning. We demonstrate through examples in these three disciplines that the unity revealed by graph representations facilitates the exchange of ideas across them.
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