Recognition of near-duplicate periodic patterns by continuous metrics with approximation guarantees

TL;DR

This paper introduces continuous metrics with approximation guarantees for recognizing near-duplicate periodic patterns in Euclidean space, addressing noise robustness and enabling efficient detection of similar crystal structures.

Contribution

It develops a novel class of continuous metrics for periodic pattern recognition that are Lipschitz continuous and computationally efficient, improving over previous finite subset descriptors.

Findings

Metrics are applicable in any Euclidean dimension.

Algorithms approximate metrics with small error bounds.

Practical application in crystal database duplicate detection.

Abstract

This paper rigorously solves the challenging problem of recognizing periodic patterns under rigid motion in Euclidean geometry. The 3-dimensional case is practically important for justifying the novelty of solid crystalline materials (periodic crystals) and for patenting medical drugs in a solid tablet form. Past descriptors based on finite subsets fail when a unit cell of a periodic pattern discontinuously changes under almost any perturbation of atoms, which is inevitable due to noise and atomic vibrations. The major problem is not only to find complete invariants (descriptors with no false negatives and no false positives for all periodic patterns) but to design efficient algorithms for distance metrics on these invariants that should continuously behave under noise. The proposed continuous metrics solve this problem in any Euclidean dimension and are algorithmically approximated with small error factors in times that are explicitly bounded in the size and complexity of a given pattern. The proved Lipschitz continuity allows us to confirm all near-duplicates filtered by simpler invariants in major databases of experimental and simulated crystals. This practical detection of noisy duplicates will stop the artificial generation of `new' materials from slight perturbations of known crystals. Several such duplicates are under investigation by five journals for data integrity.