arXiv Machine Learning
Aug 28

Gaussian Processes and Reproducing Kernel Hilbert Spaces: Connections and Equivalences

The monograph explores the relationships between Gaussian processes and reproducing kernel Hilbert spaces (RKHS), two widely used approaches that rely on positive definite kernels. It examines how these frameworks connect and are equivalent across key topics such as regression, interpolation, numerical integration, distributional discrepancies, statistical dependence, and Gaussian process sample path properties. By establishing a unifying perspective based on the equivalence between the Gaussian Hilbert space and the RKHS, the work aims to bridge methods developed independently by the machine learning, statistics, and numerical analysis communities.

By Motonobu Kanagawa, Philipp Hennig, Dino Sejdinovic, Bharath K. Sriperumbudur
arXiv Machine Learning
1d ago

A Weighted Kernel Method for Approximation that Adapts to Learned Multivariable Structure

The paper introduces Total Sensitivity Kernels (TSKs), a weighted ANOVA kernel framework that learns the importance of individual inputs and their interactions for approximating a multivariable black-box function from limited data. By selecting an RKHS where the target function has minimum norm, the authors derive a unique solution and prove consistency for finite-data interpolation. Numerical experiments show that adapting the kernel to the learned multivariable structure can significantly improve approximation accuracy compared to a standard product kernel.

By John E. Darges, Laura Weidensager
arXiv Machine Learning
2d ago

Resolution-Independent Analysis of Encoder--Decoder Operator Learning via Limiting Kernels

The paper studies operator learning on function spaces using encoder–decoder architectures. It shows that as input and output resolutions grow, the induced kernels converge to a limiting kernel, enabling regularity assumptions independent of resolution. The authors derive upper and lower bounds for regularized stochastic gradient descent, extend the analysis to neural networks via the limiting neural tangent kernel, and provide error bounds and complexity guarantees for various kernel and encoding constructions.

By Lei Shi, Jia-Qi Yang, Ding-Xuan Zhou