Operator-free Equilibrium on the Sphere

TL;DR

This paper introduces a new criterion for evaluating the uniformity of point distributions on the sphere using a generalized discrepancy derived from spherical harmonics, enabling efficient point system analysis without complex operators.

Contribution

It develops a derivative-based generalized discrepancy measure for sphere point sets, simplifying computations and enabling the discovery of minimal discrepancy point systems.

Findings

The new discrepancy measure simplifies computation using elementary functions.

Few points generated by the proposed method suffice for accurate approximation.

The method outperforms Monte Carlo in efficiency for high-dimensional sphere sampling.

Abstract

We propose a generalized minimum discrepancy, which derives from Legendre's ODE and spherical harmonic theoretics to provide a new criterion of equidistributed pointsets on the sphere. A continuous and derivative kernel in terms of elementary functions is established to simplify the computation of the generalized minimum discrepancy. We consider the deterministic point generated from Pycke's statistics to integrate a Franke function for the sphere and investigate the discrepancies of points systems embedding with different kernels. Quantitive experiments are conducted and the results are analyzed. Our deduced model can explore latent point systems, that have the minimum discrepancy without the involvement of pseudodifferential operators and Beltrami operators, by the use of derivatives. Compared to the random point generated from the Monte Carlo method, only a few points generated by our method are required to approximate the target in arbitrary dimensions.