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.

翻译：低差异设计在拟蒙特卡洛方法中占据核心地位，并日益对机器学习、机器人学和计算机图形学等其他领域产生重要影响。近年来，一种称为子集选择的低差异构造方法受到了广泛关注。给定一个大规模总体，该方法旨在基于差异度量的目标函数，最优地选出一个具有低差异性的小子集。已知该问题的若干变体是NP难的。本文首次证明，基于核差异的子集选择问题同样是NP难的。受此难解性启发，我们提出了一种利用深度嵌入核这一新概念的贝叶斯优化流程，用于解决子集选择问题。我们展示了该贝叶斯优化算法在最小化差异度量方面的性能，并指出该框架广泛适用于任何设计准则。