Thinning to improve two-sample discrepancy

TL;DR

This paper introduces an online thinning algorithm that significantly reduces the discrepancy between two samples from the same distribution, from order ( ext{n}) to polylogarithmic in n, by discarding a small fraction of points.

Contribution

The paper presents a novel online method for thinning samples to drastically decrease discrepancy, improving upon the typical ( ext{n}) order.

Findings

Discrepancy reduced from O(( ext{n})) to O(( ext{log}^{2d} n))

Algorithm is simple and online, suitable for real-time applications

Small fraction of points discarded to achieve discrepancy reduction

Abstract

The discrepancy between two independent samples \(X_1,\dots,X_n\) and \(Y_1,\dots,Y_n\) drawn from the same distribution on $\mathbb{R}^d$ typically has order \(O(\sqrt{n})\) even in one dimension. We give a simple online algorithm that reduces the discrepancy to \(O(\log^{2d} n)\) by discarding a small fraction of the points.