Geometric Data Science

TL;DR

This book introduces Geometric Data Science, focusing on classifying finite and periodic point sets with polynomial-time algorithms, and establishing a universal moduli space for crystals validated by experiments.

Contribution

It provides the first complete classification of finite point clouds and periodic crystals, developing polynomial-time invariants and a universal moduli space for crystal structures.

Findings

Complete classification of finite point clouds under rigid motion.

Hierarchy of invariants for periodic point sets.

Experimental validation of the Crystal Isometry Principle.

Abstract

This book introduces the new research area of Geometric Data Science, where data can represent any real objects through geometric measurements. The first part of the book focuses on finite point sets. The most important result is a complete and continuous classification of all finite clouds of unordered points under rigid motion in any Euclidean space. The key challenge was to avoid the exponential complexity arising from permutations of the given unordered points. For a fixed dimension of the ambient Euclidean space, the times of all algorithms for the resulting invariants and distance metrics depend polynomially on the number of points. The second part of the book advances a similar classification in the much more difficult case of periodic point sets, which model all periodic crystals at the atomic scale. The most significant result is the hierarchy of invariants from the ultra-fast to complete ones. The key challenge was to resolve the discontinuity of crystal representations that break down under almost any noise. Experimental validation on all major materials databases confirmed the Crystal Isometry Principle: any real periodic crystal has a unique location in a common moduli space of all periodic structures under rigid motion. The resulting moduli space contains all known and not yet discovered periodic crystals and hence continuously extends Mendeleev's table to the full crystal universe.