Graph Based Gaussian Processes on Restricted Domains
This addresses the issue of geometry-aware modeling for researchers in statistics and machine learning, but it is incremental as it builds on existing GP and Graph Laplacian methods.
The authors tackled the problem of sub-optimal results in nonparametric regression when inputs are in restricted Euclidean subsets by proposing Graph Laplacian based Gaussian Processes (GL-GPs), which learn a geometry-respecting covariance and show performance gains in applications.
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 Nyström type extension. We provide substantial theoretical support for the GL-GP methodology, and illustrate performance gains in various applications.