Finding well approximating lattices for a finite set of points

TL;DR

This paper develops a method using the LLL and least squares algorithms to find lattices that closely approximate a finite set of points in multiple dimensions, with proven near-optimal results in one dimension.

Contribution

It introduces a novel approach combining LLL and least squares algorithms to approximate finite point sets with lattices in multiple dimensions.

Findings

Effective lattice approximation for finite point sets

Extension of one-dimensional results to higher dimensions

Demonstrated practical examples of the approach

Abstract

In this paper we address the problem of finding well approximating lattices for a given finite set $A$ of points in ${\mathbb R}^n$. More precisely, we search for $\v{o},\v{d_1}, \dots,\v{d_n}\in \mathbb{R}^n$ such that $\v{a}-\v{o}$ is close to $\Lambda=\v{d_1}\mathbb{Z}+\dots+\v{d_n}\mathbb{Z}$ for every $\v{a}\in A$. First we deal with the one-dimensional case, where we show that in a sense the results are almost the best possible. These results easily extend to the multi-dimensional case where the directions of the axes are given, too. Thereafter we treat the general multi-dimensional case. Our method relies on the LLL algorithm. Finally we apply the least squares algorithm to optimize the results. We give several examples to illustrate our approach.