Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Lipschitz continuity of the Spherical Cap Discrepancy around generic point sets

Holger Heitsch, René Henrion

TL;DR

This work proves that the spherical cap discrepancy $\\Delta$ is locally Lipschitz around generic point sets on the sphere by leveraging an explicit finite representation formula and a differentiable extension $\\Lambda$ of the discrepancy. It introduces smooth local selections of the constituent discrepancy terms to obtain a $C^1$-based, Lipschitz surrogate, enabling computable Lipschitz constants and Clarke-based optimality analysis. The results yield a rigorous pathway to derive necessary optimality conditions for fixed-size point sets minimizing $\\Delta$ (optimal quantization) and suggest practical implications for numerical optimization on the sphere. The combination of a representation formula, a generalized discrepancy, and variational tools sharpens both the theoretical understanding and algorithmic potential for sphere-based uniformity and integration error control.

Abstract

The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a recently proven explicit formula for the spherical discrepancy, we show as a main result of this paper that this discrepancy is Lipschitz continuous in a neighbourhood of so-called generic point sets (as they are typical outcomes of Monte-Carlo sampling). This property may have some impact (both algorithmically and theoretically for deriving necessary optimality conditions) on optimal quantization, i.e., on finding point sets of fixed size on the sphere having minimum spherical discrepancy.

On the Lipschitz continuity of the Spherical Cap Discrepancy around generic point sets

TL;DR

This work proves that the spherical cap discrepancy is locally Lipschitz around generic point sets on the sphere by leveraging an explicit finite representation formula and a differentiable extension of the discrepancy. It introduces smooth local selections of the constituent discrepancy terms to obtain a -based, Lipschitz surrogate, enabling computable Lipschitz constants and Clarke-based optimality analysis. The results yield a rigorous pathway to derive necessary optimality conditions for fixed-size point sets minimizing (optimal quantization) and suggest practical implications for numerical optimization on the sphere. The combination of a representation formula, a generalized discrepancy, and variational tools sharpens both the theoretical understanding and algorithmic potential for sphere-based uniformity and integration error control.

Abstract

The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a recently proven explicit formula for the spherical discrepancy, we show as a main result of this paper that this discrepancy is Lipschitz continuous in a neighbourhood of so-called generic point sets (as they are typical outcomes of Monte-Carlo sampling). This property may have some impact (both algorithmically and theoretically for deriving necessary optimality conditions) on optimal quantization, i.e., on finding point sets of fixed size on the sphere having minimum spherical discrepancy.

Paper Structure

This paper contains 8 sections, 21 theorems, 126 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic concepts and continuity of the spherical cap discrepancy
  3. Generic point sets and a representation formula for the spherical cap discrepancy
  4. Local Lipschitz continuity of the spherical cap discrepancy at generic point sets
  5. Definition and continuity of the generalized cap discrepancy
  6. Local Lipschitz continuity of the generalized cap discrepancy at generic point sets
  7. Optimal quantization and necessary optimality conditions
  8. Appendix

Key Result

Proposition 2.1

Let $X\in\left(\mathbb{S}^{d-1}\right)^N$ be given. Then, there are $w^*\in\mathbb{S}^{d-1}$ and $t^*\in[-1,1]$ such that Moreover, there exists some $i\in\{1,\ldots ,N\}$ with $\langle w^*,x^{(i)}\rangle =t^*$.

Figures (1)

  • Figure 1: The cap discrepancy as a maximum of discontinuous elementary functions.

Theorems & Definitions (43)

  • Proposition 2.1: heitsch-henrion, Proposition 1 & 2
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • ...and 33 more