Uniform projection designs under the stratified $L_2$-discrepancy

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.