A Deformation Approach to Axially Symmetric Kernels on the Sphere: Harmonic Coupling and Regularity
Year: 2026
Type: preprint
Abstract: In this paper, we construct axially symmetric kernels on the sphere. We start by taking an isotropic kernel and transform it into an axially symmetric kernel via a specific deformation. For a general deformation, an explicit spherical-harmonic representation is not necessarily available. For our deformation, we can obtain such a representation of the deformed kernel in terms of the expansion coefficients of the original isotropic kernel. We derive this representation by explicitly describing the effect of the deformation on the spherical harmonic basis. From this, an explicit finite sum formula for the coupling coefficients is deduced and we show that, for each fixed input mode, the coefficients decay geometrically at a rate controlled by our deformation parameter. We further note that if the native space of the isotropic kernel is norm equivalent to a Sobolev-space, then the deformed kernel has the same native space with an equivalent norm. In our numerical experiments, we show that our deformation can improve prediction when the underlying random field has an axially symmetric covariance.
Source: arXiv (Cornell University)
Authors: Jäger, Janin, Stallbauer, Fabian
Institution: Catholic University of Eichstätt-Ingolstadt
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W7214239901