A Bayesian Approach to Low-Discrepancy Subset Selection

TL;DR

This paper introduces a Bayesian Optimization framework for selecting low-discrepancy subsets in high-dimensional spaces, addressing NP-hardness in discrepancy minimization and demonstrating broad applicability across design criteria.

Contribution

It establishes NP-hardness of kernel discrepancy subset selection and proposes a Bayesian Optimization method using deep embedding kernels for effective solutions.

Findings

BO effectively minimizes discrepancy measures

Framework applicable to various design criteria

Addresses NP-hardness in subset selection

Abstract

Low-discrepancy designs play a central role in quasi-Monte Carlo methods and are increasingly influential in other domains such as machine learning, robotics and computer graphics, to name a few. In recent years, one such low-discrepancy construction method called subset selection has received a lot of attention. Given a large population, one optimally selects a small low-discrepancy subset with respect to a discrepancy-based objective. Versions of this problem are known to be NP-hard. In this text, we establish, for the first time, that the subset selection problem with respect to kernel discrepancies is also NP-hard. Motivated by this intractability, we propose a Bayesian Optimization procedure for the subset selection problem utilizing the recent notion of deep embedding kernels. We demonstrate the performance of the BO algorithm to minimize discrepancy measures and note that the framework is broadly applicable any design criteria.