Graph Based Gaussian Processes on Restricted Domains
arXiv:2010.07242
Abstract
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.
arXiv comment38 pages, 15 figures
References in corpus (8)
- Embodied Hands: Modeling and Capturing Hands and Bodies Together
- Rates of contraction of posterior distributions based on Gaussian process priors
- Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth
- Reproducing kernel Hilbert spaces of Gaussian priors
- Anisotropic function estimation using multi-bandwidth Gaussian processes
- Highly Scalable Bayesian Geostatistical Modeling via Meshed Gaussian Processes on Partitioned Domains
- Mixed domain asymptotics for a stochastic process model with time trend and measurement error
- Nonparametric Bayesian Regression on Manifolds via Brownian Motion