# A Nonlocal Functional Promoting Low-Discrepancy Point Sets

arXiv: https://arxiv.org/abs/1902.00441 · HTML: https://mlsanity.com/paper/1902.00441/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Feb 2019.

## Built on by (most cited first, 1 of 1)

- [Quasi-Random Physics-informed Neural Networks](https://mlsanity.com/paper/2507.08121/) (2025)

## Abstract (arXiv)

Let X = leftx1, dots, xNright subset mathbbTd cong [0,1]d be a set of N points in the d-dimensional torus that we want to arrange as regularly possible. The purpose of this paper is to introduce a curious energy functional  E(X) = sum1 ≤ m,n ≤ N atop m neq n prodk=1d (1 - logleft(2 sin left( π |xm,k - ym,k |right) right)) and to suggest that moving a set X into the direction -nabla E(X) may have the effect of increasing regularity of the set in the sense of decreasing discrepancy. We numerically demonstrate the effect for Halton, Hammersley, Kronecker, Niederreiter and Sobol sets. Lattices in d=2 are critical points of the energy functional, some (possibly all) are strict local minima.