Optimizing Kernel Discrepancies via Subset Selection

TL;DR

This paper develops a new subset selection algorithm for kernel discrepancies, enabling efficient generation of low-discrepancy samples for various distributions, advancing quasi-Monte Carlo methods.

Contribution

It introduces a novel subset selection method applicable to general kernel discrepancies, including Stein discrepancy, for improved sample quality in QMC.

Findings

Efficient algorithm for subset selection in kernel discrepancy measures

Applicable to uniform and general distributions with known densities

Explores relationship between $L_2$ and $L_ Infty$ discrepancy measures

Abstract

Kernel discrepancies are a powerful tool for analyzing worst-case errors in quasi-Monte Carlo (QMC) methods. Building on recent advances in optimizing such discrepancy measures, we extend the subset selection problem to the setting of kernel discrepancies, selecting an m-element subset from a large population of size $n \gg m$. We introduce a novel subset selection algorithm applicable to general kernel discrepancies to efficiently generate low-discrepancy samples from both the uniform distribution on the unit hypercube, the traditional setting of classical QMC, and from more general distributions $F$ with known density functions by employing the kernel Stein discrepancy. We also explore the relationship between the classical $L_2$ star discrepancy and its $L_\infty$ counterpart.