Star Discrepancy Subset Selection: Problem Formulation and Efficient Approaches for Low Dimensions

TL;DR

This paper formulates the star discrepancy subset selection problem, proves its NP-hardness, and proposes MILP and branch-and-bound methods, showing their efficiency in low dimensions and potential for generating low-discrepancy point sets.

Contribution

It introduces the star discrepancy subset selection problem, provides formulations and algorithms, and demonstrates their effectiveness and limitations in low-dimensional settings.

Findings

MILP and BB are efficient in 2D for certain set sizes

Selected subsets have lower discrepancy than common sequences

Performance declines in higher dimensions and larger sets

Abstract

Motivated by applications in instance selection, we introduce the star discrepancy subset selection problem, which consists of finding a subset of m out of n points that minimizes the star discrepancy. First, we show that this problem is NP-hard. Then, we introduce a mixed integer linear formulation (MILP) and a combinatorial branch-and-bound (BB) algorithm for the star discrepancy subset selection problem and we evaluate both approaches against random subset selection and a greedy construction on different use-cases in dimension two and three. Our results show that the MILP and BB are efficient in dimension two for large and small $m/n$ ratio, respectively, and for not too large n. However, the performance of both approaches decays strongly for larger dimensions and set sizes. As a side effect of our empirical comparisons we obtain point sets of discrepancy values that are much smaller than those of common low-discrepancy sequences, random point sets, and of Latin Hypercube Sampling. This suggests that subset selection could be an interesting approach for generating point sets of small discrepancy value.