Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equal area partitions of the sphere with diameter bounds, via optimal transport

Jun Kitagawa, Asuka Takatsu

TL;DR

This work addresses the problem of partitioning the sphere into equal-area regions with explicit diameter control, using optimal transport both intrinsically on $\mathbb{S}^{n-1}$ and extrinsically in $\mathbb{R}^n$. The authors provide constructive partitions with diameter bounds expressed in terms of $p$-Monge–Kantorovich distances and show how these partitions bound the expected maximal diameter under uniform sampling and facilitate accurate approximations of sliced Monge–Kantorovich distances. Two OT-based constructions are developed: an intrinsic partition via a map $T_p$ pushing $\sigma_{n-1}$ to $\nu_{\vec{\omega}}$, and an extrinsic partition yielding Laguerre-type cells with diameter bounds controlled by $\mathrm{MK}_p^{\mathbb{R}^n}$. The results yield practical error bounds for sliced OT quantities, enable numerical computation through Laguerre tessellations, and extend to general spaces such as manifolds and Ahlfors regular settings, with implications for efficient spherical data analysis and OT-based metrics.

Abstract

We prove existence of equal area partitions of the unit sphere via optimal transport methods, accompanied by diameter bounds written in terms of Monge--Kantorovich distances. This can be used to obtain bounds on the expectation of the maximum diameter of partition sets, when points are uniformly sampled from the sphere. An application to the computation of sliced Monge--Kantorovich distances is also presented.

Equal area partitions of the sphere with diameter bounds, via optimal transport

TL;DR

This work addresses the problem of partitioning the sphere into equal-area regions with explicit diameter control, using optimal transport both intrinsically on and extrinsically in . The authors provide constructive partitions with diameter bounds expressed in terms of -Monge–Kantorovich distances and show how these partitions bound the expected maximal diameter under uniform sampling and facilitate accurate approximations of sliced Monge–Kantorovich distances. Two OT-based constructions are developed: an intrinsic partition via a map pushing to , and an extrinsic partition yielding Laguerre-type cells with diameter bounds controlled by . The results yield practical error bounds for sliced OT quantities, enable numerical computation through Laguerre tessellations, and extend to general spaces such as manifolds and Ahlfors regular settings, with implications for efficient spherical data analysis and OT-based metrics.

Abstract

We prove existence of equal area partitions of the unit sphere via optimal transport methods, accompanied by diameter bounds written in terms of Monge--Kantorovich distances. This can be used to obtain bounds on the expectation of the maximum diameter of partition sets, when points are uniformly sampled from the sphere. An application to the computation of sliced Monge--Kantorovich distances is also presented.
Paper Structure (3 sections, 6 theorems, 70 equations)

This paper contains 3 sections, 6 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Existence of partitions on the sphere
  3. Application to sliced Monge--Kantorovich distances

Key Result

Theorem 1.2

Let ${\vec{\omega}}=\{\omega_l\}_{l=1}^L\subset \mathbb{S}^{n-1}$ and fix $1<p<\infty$. Then there is an equal area partition $\{D_l\}_{l=1}^L$ of $\mathbb{S}^{n-1}$ such that where

Theorems & Definitions (17)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 2.1: JylhaRajala16*Theorem 1.2
  • proof : Proof of Theorem \ref{['thm: partition']}
  • Remark 2.2
  • ...and 7 more