On the $L_2$-discrepancy of Latin hypercubes

TL;DR

The paper addresses the asymptotics of the $L_2$-discrepancy for weak Latin hypercubes and uncovers a precise link between extreme and periodic discrepancies via a general energy-triple framework. It introduces weighted point-set energies, proves a central excess identity, and applies it to weak Latin hypercubes to derive explicit relations and dimension-dependent bounds. For $d\ge 3$ it provides asymptotically tight bounds, with exact constants for $d\ge 4$, and analyzes random weak Latin hypercubes to obtain expected discrepancy results that yield matching upper and lower bounds in certain regimes. The findings illuminate the limitations of weak Latin hypercubes for global optimality in high dimensions and point toward near-optimal constructions, such as Latin hypercubes built from permutations or digital nets, as promising directions for achieving low $L_2$-discrepancy.

Abstract

We investigate $L_2$-discrepancies of what we call weak Latin hypercubes. In this case it turns out that there is a precise equivalence between the extreme and periodic $L_2$-discrepancy which follows from a much broader result about generalized energies for weighted point sets. Motivated by this we study the asymptotics of the optimal $L_2$-discrepancy of weak Latin hypercubes. We determine asymptotically tight bounds for $d \geq 3$ and even the precise (dimension dependent) constant in front of the dominating term for $d \geq 4$.

Theorems & Definitions (29)

• Definition 1.1
• Theorem 1.2
• Proposition 2.1
• Definition 2.2
• Proposition 2.3
• proof : Proof of Proposition \ref{['prop:energy_triple_canonical_construction']}
• Definition 2.4
• Example 2.5
• Definition 2.6
• Lemma 2.7