Gaussian Processes and Reproducing Kernel Hilbert Spaces: Connections and Equivalences

arXiv:2506.17366 · stat.ML, cs.LG, cs.NA, math.NA, math.PR, math.ST, stat.TH · Submitted 2025-06-20 · Read on arXiv

stat.ML, cs.LG, cs.NA, math.NA, math.PR, math.ST, stat.TH

Submitted: 2025-06-20

Updated: 2026-09-09

Comments: To be published in the Institute of Mathematical Statistics Monographs series, Cambridge University Press

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Hilbert Space — 88× in this paper · explained in 2 episodes
• Gaussian Processes — 76× in this paper
• Regularization — 18× in this paper · explained in 3 episodes
• Kernel Ridge Regression — 12× in this paper · explained in 2 episodes
• Embeddings — 10× in this paper · explained in 2 episodes
• Reproducing Kernel Hilbert Spaces — 10× in this paper
• Gaussian Hilbert Spaces — 8× in this paper
• Adam — 7× in this paper · explained in 2 episodes
• Gaussian Process Regression — 7× in this paper
• Maximum Mean Discrepancy — 7× in this paper
• Uncertainty Quantification — 7× in this paper · explained in 9 episodes
• Objective Function — 5× in this paper · explained in 2 episodes
• Gaussian Process Independence Criterion — 3× in this paper
• Generalization — 3× in this paper · explained in 13 episodes
• Hilbert-Schmidt Independence Criterion — 3× in this paper
• Inverse Problem — 3× in this paper · explained in 2 episodes
• Trace — 3× in this paper · explained in 2 episodes
• Bayesian Inference — 2× in this paper · explained in 6 episodes
• Calibration — 2× in this paper · explained in 9 episodes
• Gaussian Process Discrepancy — 2× in this paper
• Bayesian Optimization — 1× in this paper · explained in 3 episodes
• Computational Biology — 1× in this paper · explained in 2 episodes
• Consistency — 1× in this paper · explained in 3 episodes
• Convergence Guarantees — 1× in this paper · explained in 3 episodes
• Curse of Dimensionality — 1× in this paper · explained in 6 episodes
• Distance Preservation — 1× in this paper · explained in 2 episodes
• Loss Function — 1× in this paper · explained in 2 episodes
• Regret — 1× in this paper · explained in 4 episodes
• Reproducing Kernel Hilbert Space (RKHS) — 1× in this paper · explained in 7 episodes
• Theoretical Guarantees — 1× in this paper · explained in 2 episodes

Sources

• A Dictionary of Closed-Form Kernel Mean Embeddings
• Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences
• Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers
• Convergence Types and Rates in Generic Karhunen-Lo\`eve Expansions with Applications to Sample Path Properties

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