Intrinsic Gaussian Processes on Complex Constrained Domains

Pokman Cheung, Lizhen Lin, Zhenwen Dai, Neil D. Lawrence, David Dunson
Journal of the Royal Statistical Society: Series B (Statistical Methodology), 81(3), 2019.

Abstract

We propose a class of intrinsic Gaussian processes (GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregularly shaped spaces arising as subsets or submanifolds of $\Re$, $\Re^2$, $\Re^3$ and beyond. For example, intrinsic GPs can accommodate spatial domains arising as complex subsets of Euclidean space. Intrinsic GPs respect the potentially complex boundary or interior conditions as well as the intrinsic geometry of the spaces. The key novelty of the approach proposed is to utilize the relationship between heat kernels and the transition density of Brownian motion on manifolds for constructing and approximating valid and computationally feasible covariance kernels. This enables intrinsic GPs to be practically applied in great generality, whereas existing approaches for smoothing on constrained domains are limited to simple special cases. The broad utilities of the intrinsic GP approach are illustrated through simulation studies and data examples.

Cite this Paper

BibTeX
@Article{intrinsic-gaussian-processes-on-complex-constrained-domains, title = {Intrinsic {G}aussian Processes on Complex Constrained Domains}, author = {Cheung, Pokman and Lin, Lizhen and Dai, Zhenwen and Lawrence, Neil D. and Dunson, David}, journal = {Journal of the Royal Statistical Society: Series B (Statistical Methodology)}, year = {2019}, volume = {81}, number = {3}, doi = {doi.org/10.1111/rssb.12320}, pdf = {https://rss.onlinelibrary.wiley.com/doi/epdf/10.1111/rssb.12320}, url = {http://inverseprobability.com/publications/intrinsic-gaussian-processes-on-complex-constrained-domains.html}, abstract = {We propose a class of intrinsic Gaussian processes (GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregularly shaped spaces arising as subsets or submanifolds of $\Re$, $\Re^2$, $\Re^3$ and beyond. For example, intrinsic GPs can accommodate spatial domains arising as complex subsets of Euclidean space. Intrinsic GPs respect the potentially complex boundary or interior conditions as well as the intrinsic geometry of the spaces. The key novelty of the approach proposed is to utilize the relationship between heat kernels and the transition density of Brownian motion on manifolds for constructing and approximating valid and computationally feasible covariance kernels. This enables intrinsic GPs to be practically applied in great generality, whereas existing approaches for smoothing on constrained domains are limited to simple special cases. The broad utilities of the intrinsic GP approach are illustrated through simulation studies and data examples. } }
Endnote
%0 Journal Article %T Intrinsic Gaussian Processes on Complex Constrained Domains %A Pokman Cheung %A Lizhen Lin %A Zhenwen Dai %A Neil D. Lawrence %A David Dunson %J Journal of the Royal Statistical Society: Series B (Statistical Methodology) %D 2019 %F intrinsic-gaussian-processes-on-complex-constrained-domains %R doi.org/10.1111/rssb.12320 %U http://inverseprobability.com/publications/intrinsic-gaussian-processes-on-complex-constrained-domains.html %V 81 %N 3 %X We propose a class of intrinsic Gaussian processes (GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregularly shaped spaces arising as subsets or submanifolds of $\Re$, $\Re^2$, $\Re^3$ and beyond. For example, intrinsic GPs can accommodate spatial domains arising as complex subsets of Euclidean space. Intrinsic GPs respect the potentially complex boundary or interior conditions as well as the intrinsic geometry of the spaces. The key novelty of the approach proposed is to utilize the relationship between heat kernels and the transition density of Brownian motion on manifolds for constructing and approximating valid and computationally feasible covariance kernels. This enables intrinsic GPs to be practically applied in great generality, whereas existing approaches for smoothing on constrained domains are limited to simple special cases. The broad utilities of the intrinsic GP approach are illustrated through simulation studies and data examples.
RIS
TY - JOUR TI - Intrinsic Gaussian Processes on Complex Constrained Domains AU - Pokman Cheung AU - Lizhen Lin AU - Zhenwen Dai AU - Neil D. Lawrence AU - David Dunson DA - 2019/04/19 ID - intrinsic-gaussian-processes-on-complex-constrained-domains VL - 81 IS - 3 DO - doi.org/10.1111/rssb.12320 L1 - https://rss.onlinelibrary.wiley.com/doi/epdf/10.1111/rssb.12320 UR - http://inverseprobability.com/publications/intrinsic-gaussian-processes-on-complex-constrained-domains.html AB - We propose a class of intrinsic Gaussian processes (GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregularly shaped spaces arising as subsets or submanifolds of $\Re$, $\Re^2$, $\Re^3$ and beyond. For example, intrinsic GPs can accommodate spatial domains arising as complex subsets of Euclidean space. Intrinsic GPs respect the potentially complex boundary or interior conditions as well as the intrinsic geometry of the spaces. The key novelty of the approach proposed is to utilize the relationship between heat kernels and the transition density of Brownian motion on manifolds for constructing and approximating valid and computationally feasible covariance kernels. This enables intrinsic GPs to be practically applied in great generality, whereas existing approaches for smoothing on constrained domains are limited to simple special cases. The broad utilities of the intrinsic GP approach are illustrated through simulation studies and data examples. ER -
APA
Cheung, P., Lin, L., Dai, Z., Lawrence, N.D. & Dunson, D.. (2019). Intrinsic Gaussian Processes on Complex Constrained Domains. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 81(3) doi:doi.org/10.1111/rssb.12320 Available from http://inverseprobability.com/publications/intrinsic-gaussian-processes-on-complex-constrained-domains.html.

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