Piecewise-linear embeddings of the space of 3D lattices into $\RR^{13}$ for high-throughput handling of lattice parameters

TL;DR

This paper introduces two piecewise-linear methods to parametrize 3D lattices in 13-dimensional space, enabling efficient comparison and enumeration of lattice structures for applications in crystallography and deep learning.

Contribution

It presents novel parametrization techniques for 3D lattices using Conway's vonorms/conorms and an extension of Ryskov's C-type, facilitating rapid lattice analysis.

Findings

Efficient lattice comparison within error margins.

Algorithm for enumerating lattice isometries under perturbations.

Application to database querying and structure generation.

Abstract

We present two methods to continuously and piecewise-linearly parametrize rank-3 lattices by vectors of $\RR^{13}$, which provides an efficient way to judge if two sets of parameters provide nearly identical lattices within their margins of errors. Such a parametrization can be used to speed up scientific computing involving periodic structures in $\RR^3$ such as crystal structures, which includes database querying, detection of duplicate entries, and structure generation via deep learning techniques. One gives a novel application of Conway's vonorms and conorms, and another is achieved through a natural extension of Ry{\u s}hkov's $C$-type to the setting modulo $3$. Voronoi vectors modulo 3 obtained in the latter approach provide an algorithm for enumerating of all potential isometries under perturbations of lattice parameters.