arXiv:2608.13497 2026 Regularization 1 ideas extracted · analyzed Sep 1, 2026
The paper gives a sharp functional on spherical fields that combines a spectral Sobolev energy with an exponential concentration term. Its transferable asset is the explicit conformally natural operator P_N, whose spherical-harmonic eigenvalues are known exactly, together with a nonnegativity guarantee at alpha >= 1/2 when the exponential density has zero center of mass. This suggests a principled regularizer for neural fields or probabilistic models defined on a sphere: suppress high-frequency oscillations and uncontrolled concentration while explicitly removing directional drift. The construction is most relevant to spherical representations, directional density models, and neural fields rather than generic Euclidean networks.
Regularize a neural scalar field on S^N with the paper's Beckner functional at the certified coefficient alpha=1/2. The loss combines a high-order spherical spectral penalty with an exponential-density term, while a center-of-mass constraint prevents the model from exploiting low-frequency directional drift.