An enumerative formula for the spherical cap discrepancy

TL;DR

This paper presents an explicit combinatorial formula for calculating the spherical cap discrepancy, enabling precise assessment of point distribution uniformity on small-dimensional spheres.

Contribution

It introduces a fully explicit, easy-to-implement enumerative formula for spherical cap discrepancy, aiding in testing sampling schemes and establishing optimality conditions.

Findings

Provides an explicit formula for small spheres and moderate samples.

Facilitates calibration and comparison of sampling schemes.

Potential to inform optimal sampling strategies.

Abstract

The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size.