Low-Discrepancy Set Post-Processing via Gradient Descent

TL;DR

This paper introduces a gradient descent-based post-processing method for low-discrepancy sets that achieves comparable uniformity results more efficiently than traditional optimization techniques, making it accessible and versatile.

Contribution

It presents a simple, efficient gradient descent approach for improving low-discrepancy sets, applicable as post-processing to existing methods for various discrepancy measures.

Findings

Gradient descent achieves comparable discrepancy reduction to expensive methods

Method is significantly more efficient and accessible

Applicable as post-processing to various low-discrepancy set generation techniques

Abstract

The construction of low-discrepancy sets, used for uniform sampling and numerical integration, has recently seen great improvements based on optimization and machine learning techniques. However, these methods are computationally expensive, often requiring days of computation or access to GPU clusters. We show that simple gradient descent-based techniques allow for comparable results when starting with a reasonably uniform point set. Not only is this method much more efficient and accessible, but it can be applied as post-processing to any low-discrepancy set generation method for a variety of standard discrepancy measures.