Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces

2019/05/26 by Ernesto De Vito, De Vito, Ernesto, Nicole Mücke +3 · 2 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Physics and Astronomy · #Cancer-related molecular mechanisms research #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Generative Adversarial Networks and Image Synthesis #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Model Reduction and Neural Networks #cs.LG #math.FA #stat.ML

paper · pdf · doi:10.48550/arxiv.1905.10913

openalex publication_date 2019/05/26 · arxiv created 2019/05/27 · arxiv updated 2019/05/28 · openalex created_date 2022/07/23 · openalex updated_date 2026/07/28

Abstract

We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples.

Citations

• Analyticity properties in scattering and spectral theory for Schrödinger operators with long-range radial potentials
• Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds
• Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities
• Some Nonlinear Problems in Riemannian Geometry

Cited by

• L²-posterior contraction rates for Gaussian process and random series priors in Bayesian nonparametric regression models
• Expressive Power of Deep Networks on Manifolds: Simultaneous Approximation

Related

• Four Properties of Reproducing Kernel Hilbert Spaces
• A Note on Reproducing Kernels for Sobolev Spaces
• Some Hilbert Spaces related with the Dirichlet space
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