In nonparametric regression, it is common for the inputs to fall in a restricted subset of Euclidean space. Typical kernel-based methods that do not take into account the intrinsic geometry of the domain across which observations are collected may produce sub-optimal results. In this article, we focus on solving this problem in the context of Gaussian process (GP) models, proposing a new class of Graph based GPs (GL-GPs), which learn a covariance that respects the geometry of the input domain. As the heat kernel is intractable computationally, we approximate the covariance using finitely-many eigenpairs of the Graph Laplacian (GL). The GL is constructed from a kernel which depends only on the Euclidean coordinates of the inputs. Hence, we can benefit from the full knowledge about the kernel to extend the covariance structure to newly arriving samples by a Nystr\"{o}m type extension. We provide substantial theoretical support for the GL-GP methodology, and illustrate performance gains in various applications.

翻译：在非参数回归中,输入通常会掉在封闭的 Euclidean 空间子集中。 基于内核的典型方法不考虑观测所收集的域的内在几何性,可能会产生亚最佳结果。在本条中,我们侧重于在高西亚进程模型的范围内解决这一问题,提出一个新的基于图形的GP(GL-GPs)类别,以图为基础的GP(GL-GPs)类别学习尊重输入域的几何学差异。由于热内核在计算上是难以控制的,我们用拉普莱奇图(GL)的有限多种电子元来估计共差。GL的GL是由一个仅依赖投入的欧克利德坐标的内核构建的。因此,我们可以从关于内核的完全了解中得益,通过Nystr\\"{o}m型扩展而将共变结构扩大到新到达的样品。我们为GL-GP方法提供了大量的理论支持,并说明了各种应用的绩效收益。