A Bayesian global Fréchet regression method is introduced via a Fréchet Bayes rule that reduces the problem to scalar tasks, allows prior-data interpolation, and remains valid under moment conditions using weak conditional expectations.
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Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences
17 Pith papers cite this work, alongside 194 external citations. Polarity classification is still indexing.
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abstract
This paper is an attempt to bridge the conceptual gaps between researchers working on the two widely used approaches based on positive definite kernels: Bayesian learning or inference using Gaussian processes on the one side, and frequentist kernel methods based on reproducing kernel Hilbert spaces on the other. It is widely known in machine learning that these two formalisms are closely related; for instance, the estimator of kernel ridge regression is identical to the posterior mean of Gaussian process regression. However, they have been studied and developed almost independently by two essentially separate communities, and this makes it difficult to seamlessly transfer results between them. Our aim is to overcome this potential difficulty. To this end, we review several old and new results and concepts from either side, and juxtapose algorithmic quantities from each framework to highlight close similarities. We also provide discussions on subtle philosophical and theoretical differences between the two approaches.
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UNVERDICTED 17roles
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A Gaussian process model enables data-adaptive integration of nonconcurrent controls in platform trials, reducing posterior variance with a non-increasing bias bound.
Mean shift interacting particle systems generate weighted samples approximating expectations under unnormalized densities by minimizing MMD through normalizing-constant-invariant dynamics.
A kernel framework over parameter space yields confidence bounds for regularized nonlinear models on adaptive data, supporting convergence analysis in Bayesian optimization.
Exponential-weight algorithm attains Õ(√(T γ_T)) adversarial regret for kernelized bandits, with matching lower bounds for SE and Matérn kernels plus a Nyström-efficient variant.
NTK neural networks achieve minimax optimal adversarial regression rates in Sobolev spaces using gradient flow with early stopping, but minimum norm interpolants are vulnerable in the overfitting regime.
Establishes a regret lower bound proving that polynomial effective optimism rules out minimax-optimal rates for GP-UCB on Matérn kernels under uniform confidence.
Hilbert space Gaussian process approximations enable closed-form IMSE acquisition functions with non-asymptotic error bounds for faster and more accurate sequential designs.
Predictively consistent priors let complex Bayesian models match or beat the out-of-sample performance of selected simpler models across linear, logistic, and nonlinear examples without explicit selection.
Gaussian process regression generalizes Green's function integration for pressure reconstruction from error-embedded gradients, matching or exceeding GFI performance on turbulence data while providing calibrated posterior uncertainty.
Kernel-based LMI reformulation of the HJB inequality with Riccati-Hessian equality constraint yields convex SDP approximations to nonlinear optimal control value functions with suboptimality bounds.
A distributed quantum Gaussian process framework is introduced for multi-agent systems with a consensus Riemannian ADMM algorithm, evaluated on elevation datasets via quantum simulator.
OSCBO adaptively balances Gaussian process sharpness and calibration in Bayesian optimization by casting hyperparameter selection as constrained online learning, while preserving sublinear regret bounds.
Proposes projected quantum kernels with misspecified GP bandit algorithms and regret bounds to trade off expressivity against learnability in quantum kernel optimization.
Active learning framework that combines D-optimality with maximin space-filling via Gaussian process surrogates to recover governing differential equations with fewer experiments than standard designs.
Bootstrap and Bayesian uncertainty estimates for ordinal embeddings from triplet data are shown to be well-calibrated in simulations.
Tutorial on Koopman operator theory, data-driven methods such as EDMD, and their use in controller design for nonlinear systems with provided simulations and code.
citing papers explorer
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Bayesian Global Fr\'echet Regression via Weak Conditional Expectations
A Bayesian global Fréchet regression method is introduced via a Fréchet Bayes rule that reduces the problem to scalar tasks, allows prior-data interpolation, and remains valid under moment conditions using weak conditional expectations.
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Robust and Data-Adaptive Integration of Nonconcurrent Data in Platform Trials via Gaussian Processes
A Gaussian process model enables data-adaptive integration of nonconcurrent controls in platform trials, reducing posterior variance with a non-increasing bias bound.
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To discretize continually: Mean shift interacting particle systems for Bayesian inference
Mean shift interacting particle systems generate weighted samples approximating expectations under unnormalized densities by minimizing MMD through normalizing-constant-invariant dynamics.
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Kernel-based guarantees for nonlinear parametric models in Bayesian optimization
A kernel framework over parameter space yields confidence bounds for regularized nonlinear models on adaptive data, supporting convergence analysis in Bayesian optimization.
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Nearly-Optimal Algorithm for Adversarial Kernelized Bandits
Exponential-weight algorithm attains Õ(√(T γ_T)) adversarial regret for kernelized bandits, with matching lower bounds for SE and Matérn kernels plus a Nyström-efficient variant.
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Adversarial Robustness of NTK Neural Networks
NTK neural networks achieve minimax optimal adversarial regression rates in Sobolev spaces using gradient flow with early stopping, but minimum norm interpolants are vulnerable in the overfitting regime.
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On the Suboptimality of GP-UCB under Polynomial Effective Optimism
Establishes a regret lower bound proving that polynomial effective optimism rules out minimax-optimal rates for GP-UCB on Matérn kernels under uniform confidence.
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Fast and Provably Accurate Sequential Designs using Hilbert Space Gaussian Processes
Hilbert space Gaussian process approximations enable closed-form IMSE acquisition functions with non-asymptotic error bounds for faster and more accurate sequential designs.
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To select or not to select: predictively consistent priors instead of model selection
Predictively consistent priors let complex Bayesian models match or beat the out-of-sample performance of selected simpler models across linear, logistic, and nonlinear examples without explicit selection.
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Pressure reconstruction from error-embedded gradient measurements: a Gaussian-process generalization of Green's function integration
Gaussian process regression generalizes Green's function integration for pressure reconstruction from error-embedded gradients, matching or exceeding GFI performance on turbulence data while providing calibrated posterior uncertainty.
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Kernel-Based LMI Approaches to Solving the Hamilton-Jacobi-Bellman Equation and Nonlinear Optimal Control
Kernel-based LMI reformulation of the HJB inequality with Riccati-Hessian equality constraint yields convex SDP approximations to nonlinear optimal control value functions with suboptimality bounds.
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Distributed Quantum Gaussian Processes for Multi-Agent Systems
A distributed quantum Gaussian process framework is introduced for multi-agent systems with a consensus Riemannian ADMM algorithm, evaluated on elevation datasets via quantum simulator.
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Online Sharp-Calibrated Bayesian Optimization
OSCBO adaptively balances Gaussian process sharpness and calibration in Bayesian optimization by casting hyperparameter selection as constrained online learning, while preserving sublinear regret bounds.
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Balancing Expressivity and Learnability in Quantum Kernel Bandit Optimization
Proposes projected quantum kernels with misspecified GP bandit algorithms and regret bounds to trade off expressivity against learnability in quantum kernel optimization.
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Gaussian Process Assisted Active Learning of Physical Laws
Active learning framework that combines D-optimality with maximin space-filling via Gaussian process surrogates to recover governing differential equations with fewer experiments than standard designs.
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Uncertainty Estimates for Ordinal Embeddings
Bootstrap and Bayesian uncertainty estimates for ordinal embeddings from triplet data are shown to be well-calibrated in simulations.
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Koopman operator theory: fundamentals, control, and applications
Tutorial on Koopman operator theory, data-driven methods such as EDMD, and their use in controller design for nonlinear systems with provided simulations and code.