# Sobolev Spaces, Kernels and Discrepancies over Hyperspheres

arXiv: https://arxiv.org/abs/2211.09196 · HTML: https://mlsanity.com/paper/2211.09196/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 48 references that are in the map.
- First posted on arXiv 16 Nov 2022.

## Builds on (most cited first, 8 of 8)

- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Measuring Sample Quality with Stein's Method](https://mlsanity.com/paper/1506.03039/) (2015)
- [Spatial Deep Learning for Wireless Scheduling](https://mlsanity.com/paper/1808.01486/) (2018)
- [Optimal quantisation of probability measures using maximum mean discrepancy](https://mlsanity.com/paper/2010.07064/) (2020)
- [Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2110.11847/) (2021)
- [A Fourier representation of kernel Stein discrepancy with application to Goodness-of-Fit tests for measures on infinite dimensional Hilbert spaces](https://mlsanity.com/paper/2206.04552/) (2022)

## Built on by (most cited first, 6 of 6)

- [Mind the spikes: Benign overfitting of kernels and neural networks in fixed dimension](https://mlsanity.com/paper/2305.14077/) (2023)
- [A Comprehensive Analysis on the Learning Curve in Kernel Ridge Regression](https://mlsanity.com/paper/2410.17796/) (2024)
- [Posterior Contraction Rates for Mat'ern Gaussian Processes on Riemannian Manifolds](https://mlsanity.com/paper/2309.10918/) (2023)
- [On the Impacts of the Random Initialization in the Neural Tangent Kernel Theory](https://mlsanity.com/paper/2410.05626/) (2024)
- [Beyond ReLU: How Activations Affect Neural Kernels and Random Wide Networks](https://mlsanity.com/paper/2506.22429/) (2025)
- [Optimal Confidence Band for Kernel Gradient Flow Estimator](https://mlsanity.com/paper/2605.05768/) (2026)

## Abstract (arXiv)

This work provides theoretical foundations for kernel methods in the hyperspherical context. Specifically, we characterise the native spaces (reproducing kernel Hilbert spaces) and the Sobolev spaces associated with kernels defined over hyperspheres. Our results have direct consequences for kernel cubature, determining the rate of convergence of the worst case error, and expanding the applicability of cubature algorithms based on Stein's method. We first introduce a suitable characterisation on Sobolev spaces on the d-dimensional hypersphere embedded in (d+1)-dimensional Euclidean spaces. Our characterisation is based on the Fourier--Schoenberg sequences associated with a given kernel. Such sequences are hard (if not impossible) to compute analytically on d-dimensional spheres, but often feasible over Hilbert spheres. We circumvent this problem by finding a projection operator that allows to Fourier mapping from Hilbert into finite dimensional hyperspheres. We illustrate our findings through some parametric families of kernels.