Weighted $L_p$-Discrepancy Bounds for Parametric Stratified Sampling and Applications to High-Dimensional Integration

TL;DR

This paper develops bounds for the expected weighted $L_p$-discrepancy of parametric stratified sampling schemes, providing theoretical guarantees and demonstrating improved high-dimensional integration performance over classical methods.

Contribution

It introduces a parametric family of stratified partitions and derives explicit bounds for weighted $L_p$-discrepancy, advancing the theoretical understanding of stratified sampling in high dimensions.

Findings

Explicit bounds for $p=2$ using exact formulas

Probabilistic bounds for $p>2$ via dyadic chaining

Improved integration accuracy over jittered sampling

Abstract

This paper studies the expected $L_p$-discrepancy ($2 \leq p < \infty$) for stratified sampling schemes under importance sampling. We introduce a parametric family of equivolume partitions $\Omega_{\theta,\sim}$ and leverage recent exact formulas for the expected $L_2$-discrepancy \cite{xian2025improved}. Our main contribution is a weighted discrepancy reduction lemma that relates weighted $L_p$-discrepancy to standard $L_p$-discrepancy with explicit constants depending on the weight function. For $p=2$, we obtain explicit bounds using the exact discrepancy formulas. For $p>2$, we derive probabilistic bounds via dyadic chaining techniques. The results yield uniform error estimates for multivariate integration in Sobolev spaces $\mathcal{H}^1(K)$ and $F^*_{d,q}$, demonstrating improved performance over classical jittered sampling in importance sampling scenarios. Numerical experiments validate our theoretical findings and illustrate the practical advantages of parametric stratified sampling.