# Graph Based Gaussian Processes on Restricted Domains

arXiv: https://arxiv.org/abs/2010.07242 · HTML: https://mlsanity.com/paper/2010.07242/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2020.

## Built on by (most cited first, 10 of 10)

- [The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphs](https://mlsanity.com/paper/2407.08086/) (2024)
- [Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach](https://mlsanity.com/paper/2203.00126/) (2022)
- [Optimization on Manifolds via Graph Gaussian Processes](https://mlsanity.com/paper/2210.10962/) (2022)
- [Posterior Contraction Rates for Mat'ern Gaussian Processes on Riemannian Manifolds](https://mlsanity.com/paper/2309.10918/) (2023)
- [Implicit Manifold Gaussian Process Regression](https://mlsanity.com/paper/2310.19390/) (2023)
- [Hierarchical shrinkage Gaussian processes: applications to computer code emulation and dynamical system recovery](https://mlsanity.com/paper/2302.00755/) (2023)
- [Nonparametric regression on random geometric graphs sampled from submanifolds](https://mlsanity.com/paper/2405.20909/) (2024)
- [Boundary Detection Algorithm Inspired by Locally Linear Embedding](https://mlsanity.com/paper/2406.18456/) (2024)
- [Intrinsic Gaussian Process Regression Modeling for Manifold-valued Response Variable](https://mlsanity.com/paper/2411.18989/) (2024)
- [Low-rank computation of the posterior mean in Multi-Output Gaussian Processes](https://mlsanity.com/paper/2504.21527/) (2025)

## Abstract (arXiv)

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