# Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces

arXiv: https://arxiv.org/abs/1905.10913 · HTML: https://mlsanity.com/paper/1905.10913/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2019.

## Built on by (most cited first, 14 of 14)

- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels](https://mlsanity.com/paper/2111.01460/) (2021)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spaces](https://mlsanity.com/paper/2301.13088/) (2023)
- [Spectral Representation Learning for Conditional Moment Models](https://mlsanity.com/paper/2210.16525/) (2022)
- [Intrinsic Gaussian Vector Fields on Manifolds](https://mlsanity.com/paper/2310.18824/) (2023)
- [Posterior Contraction Rates for Mat'ern Gaussian Processes on Riemannian Manifolds](https://mlsanity.com/paper/2309.10918/) (2023)
- [On the speed of uniform convergence in Mercer's theorem](https://mlsanity.com/paper/2205.00487/) (2022)
- [Implicit Manifold Gaussian Process Regression](https://mlsanity.com/paper/2310.19390/) (2023)
- [Residual Deep Gaussian Processes on Manifolds](https://mlsanity.com/paper/2411.00161/) (2024)
- [Stochastic Poisson Surface Reconstruction with One Solve using Geometric Gaussian Processes](https://mlsanity.com/paper/2503.19136/) (2025)
- [Expressive Power of Deep Networks on Manifolds: Simultaneous Approximation](https://mlsanity.com/paper/2509.09362/) (2025)
- [Spectral Algorithms on Manifolds through Diffusion](https://mlsanity.com/paper/2403.03669/) (2024)
- [Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems](https://mlsanity.com/paper/2605.04126/) (2026)
- [Diffusion-based Semi-supervised Spectral Algorithm for Regression on Manifolds](https://mlsanity.com/paper/2410.14539/) (2024)

## Abstract (arXiv)

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.