On Combinatorial Properties of Greedy Wasserstein Minimization

TL;DR

This paper explores the combinatorial regularity of sequences generated by a greedy Wasserstein minimization process, revealing new regularity properties and connections to combinatorics and number theory.

Contribution

It introduces a novel greedy sequence construction based on Wasserstein distance minimization and proves a regularity result below the square root barrier.

Findings

Sequences exhibit remarkable combinatorial regularity.

Sequences match those introduced by Kritzinger in different contexts.

Numerical regularity rivals best known combinatorial constructions.

Abstract

We discuss a phenomenon where Optimal Transport leads to a remarkable amount of combinatorial regularity. Consider infinite sequences $(x_k)_{k=1}^{\infty}$ in $[0,1]$ constructed in a greedy manner: given $x_1, \dots, x_n$, the new point $x_{n+1}$ is chosen so as to minimize the Wasserstein distance $W_2$ between the empirical measure of the $n+1$ points and the Lebesgue measure, $$x_{n+1} = \arg\min_x ~W_2\left( \frac{1}{n+1} \sum_{k=1}^{n} \delta_{x_k} + \frac{\delta_{x}}{n+1}, dx\right).$$ This leads to fascinating sequences (for example: $x_{n+1} = (2k+1)/(2n+2)$ for some $k \in \mathbb{Z}$) which coincide with sequences recently introduced by Ralph Kritzinger in a different setting. Numerically, the regularity of these sequences rival the best known constructions from Combinatorics or Number Theory. We prove a regularity result below the square root barrier.