Low-Rank Neural Regression Models for Spatial Data on Complex Domains
针对非欧几里得域上的复杂空间数据，提出低秩空间回归模型，利用图拉普拉斯和拉普拉斯-贝尔特拉米算子的特征函数进行谱表示，并扩展至非线性神经回归模型，通过四个应用展示其处理不规则形状域和流形结构的能力。
Complex spatial datasets defined on non-Euclidean domains with intricate geometry and dependence structures are fostering new frontiers in statistics. For such data, traditional spatial models developed for Euclidean settings may be inadequate, motivating approaches that explicitly account for the underlying domain geometry and locally varying dependencies. In this paper, we address the modeling of random functions over broad and complex domains by adopting low-rank spatial regression models. The underlying spatial field is represented in a reduced-dimensional subspace, obtained through a basis expansion in the corresponding functional space. These models leverage the prototypical structures of manifolds and graphs, allowing for spectral representations of the data using eigenfunctions of the graph Laplacian and the Laplace-Beltrami operator. This framework is extended to non-linear spatial regression models using a feed-forward neural networks, which can effectively handle non-Euclidean domains with complex geometries, dependencies, and manifold structures. We illustrate the proposed approach through four applications, each involving distinct types of data with increasing levels of complexity in terms of spatial domain and process characteristics. The results demonstrate the usefulness of the proposed models in accommodating intricacies of irregularly-shaped spatial domains or manifold structures.