Uniform projection designs under the stratified $L_2$-discrepancy

TL;DR

This paper introduces a new uniform projection criterion, $\

Contribution

It derives explicit formulas and bounds for the criterion, linking it to known optimal designs and demonstrating its computational efficiency and effectiveness.

Findings

Many optimal designs attain the lower bound of $\

paper_type":"method","github_repo":null}}# Explanation: The abstract discusses a new criterion for space-filling designs, provides formulas, bounds, and computational aspects, and illustrates its effectiveness, fitting the 'method' category. No GitHub URL is mentioned.}# Answer: {

contribution":"It derives explicit formulas and bounds for the criterion, linking it to known optimal designs and demonstrating its computational efficiency and effectiveness.",

Abstract

This paper studies a uniform projection criterion for space-filling designs under the stratified $L_2$-discrepancy. The criterion, denoted by $\Phi_{SD}$, is the average squared stratified $L_2$-discrepancy over all two-dimensional projections. For U-type $(n,m,s^p)$ designs, we derive an explicit formula for $\Phi_{SD}$ in terms of row-pairwise weighted hierarchical distances, and we establish sharp lower and upper bounds with equality conditions. We further show that many known optimal constructions attain the lower bound of $\Phi_{SD}$, and that designs attaining the lower bound of the full stratified $L_2$-discrepancy also attain the lower bound of $\Phi_{SD}$. The criterion can be evaluated in $O(n^2m)$ time, with a modest reduction in arithmetic operations compared with direct projection-wise evaluation. Numerical studies illustrate the theoretical results and show that $\Phi_{SD}$ is effective for assessing low-dimensional projection uniformity.