Graph based Gaussian processes on restricted domains

• 成果类型: Article
• 署名作者: Dunson, David B.; Wu, Hau-Tieng; Wu, Nan
• 署名单位: Duke University; Duke University
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1111/rssb.12486
• 发表日期: 2022
• 页码: 414-439
• 关键词: CONVERGENCE-RATES; PROCESS MODELS; regression; laplacian

摘要

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 Laplacian 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 Nystrom type extension. We provide substantial theoretical support for the GL-GP methodology, and illustrate performance gains in various applications.