arXiv subjects

Daniel Widdowson

Publications and source records attributed to Daniel Widdowson.

5 recordsLinked to original sources

arXiv2025

Continuous invariant-based asymmetries of periodic crystals quantify deviations from higher symmetry

Ideal symmetry is known to break down under almost any noise. One measure of asymmetry in a periodic crystal is the relative multiplicity Z' of geometrically non-equivalent units. However, Z' discontinuously changes under almost any displacement of atoms, which can arbitrarily scale up a primitive cell. This discontinuity was recently resolved by a hierarchy of invariant descriptors that continuously change under all small perturbations. We introduce a Continuous Invariant-based Asymmetry (CIA) to quantify (in physically meaningful Angstroms) the deviation of a periodic crystal from a higher symmetry form. Our experiments on several Crystal Structure Prediction datasets show that about a half of simulated crystals have high values of CIA, while all experimental structures in these datasets have CIA=0. On another hand, many crystals with high values Z' in the Cambridge Structural Database (CSD) turned out to be close to more symmetric forms with Z'<=1 due to low values of CIAs.

math.MG↗

arXiv2023

Recognizing Rigid Patterns of Unlabeled Point Clouds by Complete and Continuous Isometry Invariants with no False Negatives and no False Positives

Rigid structures such as cars or any other solid objects are often represented by finite clouds of unlabeled points. The most natural equivalence on these point clouds is rigid motion or isometry maintaining all inter-point distances. Rigid patterns of point clouds can be reliably compared only by complete isometry invariants that can also be called equivariant descriptors without false negatives (isometric clouds having different descriptions) and without false positives (non-isometric clouds with the same description). Noise and motion in data motivate a search for invariants that are continuous under perturbations of points in a suitable metric. We propose the first continuous and complete invariant of unlabeled clouds in any Euclidean space. For a fixed dimension, the new metric for this invariant is computable in a polynomial time in the number of points.

cs.CV↗

arXiv2022

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

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.

math.MG↗

arXiv2021

Pointwise Distance Distributions for detecting near-duplicates in large materials databases

Many real objects are modeled as discrete sets of points, such as corners or other salient features. For our main applications in chemistry, points represent atomic centers in a molecule or a solid material. We study the problem of classifying discrete (finite and periodic) sets of unordered points under isometry, which is any transformation preserving distances in a metric space. Experimental noise motivates the new practical requirement to make such invariants Lipschitz continuous so that perturbing every point in its epsilon-neighborhood changes the invariant up to a constant multiple of epsilon in a suitable distance satisfying all metric axioms. Since the given points are unordered, the key challenge is to compute all invariants and metrics in a near-linear time of the input size. We define the Pointwise Distance Distribution (PDD) for any discrete set and prove, in addition to the properties above, the completeness of PDD for all periodic sets in general position. The PDD can compare nearly 2 million crystals from the world's five largest databases within 2 hours on a modest desktop computer. The impact is upholding data integrity in crystallography because the PDD will not allow anyone to claim a `new' material as a noisy disguise of a known crystal.

cs.CG↗

arXiv2020

The asymptotic behaviour and a near linear time algorithm for isometry invariants of periodic sets

The fundamental model of a periodic structure is a periodic point set up to rigid motion or isometry. Our recent paper in SoCG 2021 defined isometry invariants (density functions), which are complete in general position and continuous under perturbations. This work introduces much faster isometry invariants (average minimum distances), which are also continuous and distinguish some sets that have identical density functions. We explicitly describe the asymptotic behaviour of the new invariants for a wide class of sets including non-periodic. The proposed near linear time algorithm processed a dataset of hundreds of thousands of real structures in a few hours on a modest desktop.

cond-mat.mtrl-sci↗