Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces
This work provides theoretical insights into kernel methods on manifolds, which is incremental for mathematical analysis and machine learning applications.
The authors investigated reproducing kernel Hilbert spaces (RKHS) on Riemannian manifolds, establishing conditions for Sobolev spaces to be RKHS and characterizing their kernels, while introducing diffusion spaces as a smoother class, with results demonstrated through examples.
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.