Non-Abelian Analogs of Lattice Rounding

TL;DR

This paper explores non-abelian lattice rounding by developing algorithms for matrix groups and establishing inapproximability results, advancing understanding of computational complexity in non-commutative settings.

Contribution

It introduces algorithms for normed word problems in matrix groups and proves inapproximability results, extending lattice rounding concepts to non-abelian groups.

Findings

Algorithm successfully solves normed word problems for random matrix products

Theoretical justification supports the algorithm's effectiveness

Inapproximability results show limitations of approximation algorithms in this setting

Abstract

Lattice rounding in Euclidean space can be viewed as finding the nearest point in the orbit of an action by a discrete group, relative to the norm inherited from the ambient space. Using this point of view, we initiate the study of non-abelian analogs of lattice rounding involving matrix groups. In one direction, we give an algorithm for solving a normed word problem when the inputs are random products over a basis set, and give theoretical justification for its success. In another direction, we prove a general inapproximability result which essentially rules out strong approximation algorithms (i.e., whose approximation factors depend only on dimension) analogous to LLL in the general case.