Learning Gaussian processes which map functions to functions
December 7, 2020 — February 25, 2022
Gaussian
generative
geometry
Hilbert space
how do science
kernel tricks
machine learning
PDEs
physics
regression
spatial
stochastic processes
time series
In which I discover how to learn operators via GPs. I suspect a lot of things break; What is a usable gaussian distribution over a mapping between functions?
It might be handy here to revisit the notation for Bayesian nonparametrics, since we don’t get the same kind of setup as when the distributions in question are finitely parameterised. TBC
1 Universal Kriging
Does universal kriging fit in this notebook? (Menafoglio, Secchi, and Rosa 2013) In this setting our observations are function-valued and we wish to spatially interpolate them. TBC. Keyword: Hilbert-Kriging. See Júlio Hoffimann’s Hilbert-Kriging lecture.
2 Hilbert-space valued GPs
TBC
3 References
Bakka, Rue, Fuglstad, et al. 2018. “Spatial Modeling with R-INLA: A Review.” WIREs Computational Statistics.
Bolin. 2016. Models and Methods for Random Fields in Spatial Statistics with Computational Efficiency from Markov Properties.
Brault, d’Alché-Buc, and Heinonen. 2016. “Random Fourier Features for Operator-Valued Kernels.” In Proceedings of The 8th Asian Conference on Machine Learning.
Brault, Lim, and d’Alché-Buc. n.d. “Scaling up Vector Autoregressive Models With Operator-Valued Random Fourier Features.”
Brouard, Szafranski, and D’Alché-Buc. 2016. “Input Output Kernel Regression: Supervised and Semi-Supervised Structured Output Prediction with Operator-Valued Kernels.” The Journal of Machine Learning Research.
Davison, and Ortiz. 2019. “FutureMapping 2: Gaussian Belief Propagation for Spatial AI.” arXiv:1910.14139 [Cs].
Dutordoir, Saul, Ghahramani, et al. 2022. “Neural Diffusion Processes.”
Gahungu, Lanyon, Álvarez, et al. 2022. “Adjoint-Aided Inference of Gaussian Process Driven Differential Equations.” In.
Heinonen, and d’Alché-Buc. 2014. “Learning Nonparametric Differential Equations with Operator-Valued Kernels and Gradient Matching.” arXiv:1411.5172 [Cs, Stat].
Hennig, Osborne, and Girolami. 2015. “Probabilistic Numerics and Uncertainty in Computations.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.
Hildeman, Bolin, and Rychlik. 2019. “Joint Spatial Modeling of Significant Wave Height and Wave Period Using the SPDE Approach.” arXiv:1906.00286 [Stat].
Hu, and Steinsland. 2016. “Spatial Modeling with System of Stochastic Partial Differential Equations.” WIREs Computational Statistics.
Kadri, Duflos, Preux, et al. 2016. “Operator-Valued Kernels for Learning from Functional Response Data.” The Journal of Machine Learning Research.
Kadri, Rakotomamonjy, Preux, et al. 2012. “Multiple Operator-Valued Kernel Learning.” Advances in Neural Information Processing Systems.
Lian. 2007. “Nonlinear Functional Models for Functional Responses in Reproducing Kernel Hilbert Spaces.” Canadian Journal of Statistics.
Lim, d’Alché-Buc, Auliac, et al. 2015. “Operator-Valued Kernel-Based Vector Autoregressive Models for Network Inference.” Machine Learning.
Long, Wang, Krishnapriyan, et al. 2022. “AutoIP: A United Framework to Integrate Physics into Gaussian Processes.”
Menafoglio, Secchi, and Rosa. 2013. “A Universal Kriging Predictor for Spatially Dependent Functional Data of a Hilbert Space.” Electronic Journal of Statistics.
Minh. 2022. “Finite Sample Approximations of Exact and Entropic Wasserstein Distances Between Covariance Operators and Gaussian Processes.” SIAM/ASA Journal on Uncertainty Quantification.
Phillips, Seror, Hutchinson, et al. 2022. “Spectral Diffusion Processes.” In.
Saha, and Balamurugan. 2020. “Learning with Operator-Valued Kernels in Reproducing Kernel Krein Spaces.” In Advances in Neural Information Processing Systems.
Zammit-Mangion, and Cressie. 2021. “FRK: An R Package for Spatial and Spatio-Temporal Prediction with Large Datasets.” Journal of Statistical Software.
Zhang, Xu, and Zhang. 2012. “Refinement of Operator-Valued Reproducing Kernels.” The Journal of Machine Learning Research.