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

Olga Anosova, Daniel Widdowson, Vitaliy Kurlin

TL;DR

The paper tackles the problem of recognizing near-duplicate periodic patterns under rigid motion in Euclidean space by introducing a boundary-tolerant local metric $BT$ and extending to a complete invariant via Earth Mover's Distance on isosets. It proves Lipschitz continuity of the resulting distance and provides polynomial-time approximation guarantees, enabling robust comparison of periodic point sets in any dimension. The core result is a complete isometry classification via isosets: two periodic sets are isometric iff their isosets at a common stable radius are bijectively equivalent with matching weights. Applications to crystal databases (CSD, GNoME) demonstrate practical utility for detecting near-duplicates and ensuring data integrity, with mirror-image distinctions achievable where prior descriptors fail.

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.

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

TL;DR

The paper tackles the problem of recognizing near-duplicate periodic patterns under rigid motion in Euclidean space by introducing a boundary-tolerant local metric and extending to a complete invariant via Earth Mover's Distance on isosets. It proves Lipschitz continuity of the resulting distance and provides polynomial-time approximation guarantees, enabling robust comparison of periodic point sets in any dimension. The core result is a complete isometry classification via isosets: two periodic sets are isometric iff their isosets at a common stable radius are bijectively equivalent with matching weights. Applications to crystal databases (CSD, GNoME) demonstrate practical utility for detecting near-duplicates and ensuring data integrity, with mirror-image distinctions achievable where prior descriptors fail.

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.
Paper Structure (8 sections, 23 theorems, 7 equations, 15 figures, 2 tables, 2 algorithms)

This paper contains 8 sections, 23 theorems, 7 equations, 15 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Continuous metric problem for periodic point sets and crystals
  2. Past work on distances and invariants of periodic point sets
  3. Isometry classification of periodic point sets by complete invariants
  4. Continuous metrics on isometry classes of periodic sets in $\mathbb R^n$
  5. Algorithms to test isometry and to approximate metrics on isosets
  6. A lower bound for continuous metrics via simpler invariants
  7. Real experiments, limitations, significance, and a discussion
  8. Appendix A: detailed proofs of all auxiliary results

Key Result

Lemma 3.6

(a) Let $S\subset\mathbb R^n$ be a periodic point set with a unit cell $U$, which has the longest edge $b$ and longest diagonal $d$. Set $r(U)=\max\{b,\frac{d}{2}\}$. Then the bridge length $\beta(S)$ from Definition dfn:bridge_length has the upper bound $\min\{2R(S),r(U)\}\geq\beta(S)$. (b) For any

Figures (15)

  • Figure 1: Left: three (of infinitely many) primitive cells $U,U',U"$ of the same minimal area for the hexagonal lattice $\Lambda$. Other images show periodic sets $\Lambda+M$ with different cells and motifs, which are all isometric to $\Lambda$ whose hexagonal Voronoi domain is highlighted in yellow.
  • Figure 2: Past descriptors based on a primitive cell cannot continuously quantify a distance between near-duplicates. For example, under almost any perturbation, symmetry groups break down and a primitive cell volume discontinuously changes up to any integer factor.
  • Figure 3: Left: for any lattice $S$ and a fixed size of a box or a ball, one can choose many non-isometric finite subsets of different sizes. Right: the blue set $S$ and green set $Q$ in the line $\mathbb R$ have a small Hausdorff distance $d_H=\varepsilon$ but are not related by a small perturbation.
  • Figure 4: Left: the 1-dimensional set $S_4=\{0,\frac{1}{4},\frac{1}{3},\frac{1}{2}\}+\mathbb Z$ has four points in the unit cell $[0,1)$ and is 4-regular by Definition \ref{['dfn:m-regular']}. Right: the colored disks show $\alpha$-clusters in the line $\mathbb R$ with radii $\alpha=0,\frac{1}{12},\frac{1}{6},\frac{1}{4},\frac{3}{4}$ and represent points in the isotree $\mathrm{IT}(S_4)$ from Definition \ref{['dfn:isotree']}.
  • Figure 5: The isotree of any lattice $\Lambda$ is $[0,+\infty)$ is a line $\mathbb R$ parametrized by the radius $\alpha$. Left: the isotree of the hexagonal lattice $\Lambda_6$. Right: the isotree of the square lattice $\Lambda_4$.
  • ...and 10 more figures

Theorems & Definitions (68)

  • Definition 1.1: metric
  • Definition 2.1: Hausdorff distance $d_H$, bottleneck distance $d_B$
  • Definition 3.1: global clusters and $m$-regular periodic sets
  • Definition 3.2: local $\alpha$-clusters $C(S,p;\alpha)$ and symmetry groups $\mathrm{Sym}(S,p;\alpha)$
  • Definition 3.3: bridge length $\beta(S)$
  • Definition 3.4: isotree $\mathrm{IT}(S)$ of $\alpha$-partitions
  • Definition 3.5: the minimum stable radius $\alpha(S)$
  • Lemma 3.6: upper bounds for a stable radius $\alpha(S)$ and bridge length $\beta(S)$
  • proof
  • Example 3.7: upper bounds for $\alpha(S)$ and $\beta(S)$
  • ...and 58 more