We describe an efficient algorithm to compute the bridge length estimating the size of a complete isoset invariant, which classifies all periodic point sets under Euclidean motion.

Prof Vitaliy Kurlin: news, research, papers, teaching

University of Liverpool

arXiv abstract link

**Authors:** Olga D Anosova, Vitaliy A Kurlin 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.

**Authors:** Daniel E Widdowson, Vitaliy A Kurlin Periodic point sets model all solid crystalline materials (crystals) whose atoms can be considered zero-sized points with or without atomic types. This paper addresses the fundamental problem of checking whether claimed crystals are novel, not noisy perturbations of known materials obtained by unrealistic atomic replacements. Such near-duplicates have already skewed ground truth because past comparisons relied on discontinuous cells and symmetries. The proposed Lipschitz continuity under noise is a new essential requirement for machine learning on any data objects that have ambiguous representations and live in continuous spaces. For periodic point sets under isometry (any distance-preserving transformation), we designed invariants that distinguish all known counter-examples to the completeness of past descriptors and confirm thousands of (near-)duplicates in the world's largest databases of inorganic crystals within hours on a desktop computer.