Higher-order, generically complete, continuous, and polynomial-time isometry invariants of periodic sets

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 skewed ground-truth because past comparisons relied on unstable 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 detect thousands of (near-)duplicates in large high-profile databases of crystals within two days on a modest desktop computer.