Resolving the data ambiguity for periodic crystals
Authors: Daniel Widdowson
Vitaliy Kurlin
Venue: NeurIPS 2022 Accept (2022)
Keywords: computational geometry
data ambiguity
materials applications

Abstract: The fundamental model of all solid crystalline materials is a periodic set of atomic centers considered up to rigid motion in Euclidean space. The major obstacle to materials discovery was highly ambiguous representations of periodic crystals that didn't allow fast and reliable comparisons and led to numerous (near-) duplicates in many databases of experimental and simulated crystals. This paper exemplarily resolves the ambiguity by invariants, which are descriptors without false negatives.

The new Pointwise Distance Distributions (PDD) is a numerical matrix with a near-linear time complexity and an exactly computable metric. The strongest theoretical result is generic completeness (absence of false positives) for all finite and periodic sets of points in any dimension. The strength of PDD is shown by 200B+ pairwise comparisons of all periodic structures in the world's largest collection (Cambridge Structural Database) of existing materials over two days on a modest desktop.


Resolving the data ambiguity for periodic crystals
Daniel E Widdowson
Vitaliy A Kurlin
Department Computer Science University of Liverpool, Liverpool, L69 3BX D.E. W.W.Widwwwwoo@ verpooo.........eooooooooollo
Department of Computer Science University of Liverpool, Liverpool, L69 3BX vitaliy. kurl @nrmainlmmllll com
Abstract
The fundamental model of all solid crystalline materials is periodic set of atomic centers considered up to rigid motion in Euclidean space. The major obstacle to materials discovery was highly ambiguous representations of periodic crystals that didn't allow fast and reliable comparisons and to numerous (near-) duplicates in many databases of experimental and simulated crystals. This paper exemplarily resolves the ambiguity by invariants, which are descriptors without false negatives. The new Pointwise Distance Distributions (PDD) is numerical matrix with near- linear time complexity and exactly computable metric. The strongest theoretical result is generic completeness (absence of false positives) for all finite and periodic of points in any dimension. The strength of PDD shown by 200B+ pairwise comparisons of all periodic structures in the world's largest collection (Cambridge Structural Database) of existing materials over two days on modest desktop.
Motivations for resolving the ambiguity challenge for periodic crystals
This paper resolves the long-standing challenge of ambiguous data representation for periodic structures that model all solid crystalline materials (periodic crystals). Any real crystal is best modeled as periodic set S C IR of points at all atomic centers, whose positions have a physical meaning and are determined via diffraction patterns. Edges between points are excluded because they only abstractly represent inter-atomic bonds depending on thresholds for distances and angles (29
The simplest example is lattice A C IR" consisting of all integer linear combinations of basis whose vectors span a unit cell U. Fig. shows omly a few of infinitely many possible choices of U.
Figure 1: These isometric lattices are given by different cells and motifs. 1st: U ((1,0) 1))
M [(0,0)). 2nd: U (11,0), (0, 1)), M = ( 1). 3rd: U ((1,0), (1, 1)), M = ().
1 M - ((),0)) 4th: U = (1 1/ ) vz 1/2 M (G 2)). 5th: ((V2.0), VI
Materials discovery still relies on trial-and-errol because periodic crystals are traditionally represented by non-invariants (descriptors with false negatives) or discontinuous invariants such as symmetry groups that break down under tiny perturbations. These conventional descriptions cannot identify fraudulent structures in experimental datasets that keep depositing numerous (near-) duplicates without reliable tools for justified comparisons (30]. The ambiguity challenge will rigorously stated in Problem
classification of periodic sets up to isometry preserving the crystal rigidity.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

---

Fig. Iillutraate the first obstacle: the same lattice can be generated by infinitely many different bases or cells, so distinguishing only lattices up isometry already non-trivial. Any periodic point ser S can be defined as a sum A + M {ū p points in the basis of U. Any lattice A is considered as a periodic set with a 1-point motif M = {p)- A single point can be arbitrarily chosen in a unit cell U, see the first two pictures of Fig. [ Basis vectors of U and atomic coordinates of motif points (atomic centers) M constitute the data found in conventional Crystallographir Information File (CIF). Fig. 2 (left) shows the ambiguity of the CIF pair (A, M) even if basis is fixed. The recent work by Widdowson et [67] rigorously stated the problem continuosly classify periodic point sets up to isometry. An isometry of Euclidean space I" is any map that maintains inter-point distances. Any iiientation-preeevveee realized as continuous rigid motion, for example any composition of translations and rotations in RR3. This equivalence is most natural for finite and periodic sets that represent real rigid structures.
A, E M). where motif M is finite set of
isometry can be
the same square lattice
very small perturbation changes minimal cell
motif of points + square lattice
what isometry invariants can continuosuly quantify small differences between these periodic point sets?
periodic point set another motif
Figure 2: Left: : even for fixed cell of a lattice A, different motifs M can define isometric periodic
sets A M. Right: for almost any perturbation, the symmetry group and (the minimum volume of) any reduced cell discontinuously change, which justifies continuity condition (T.Td) in Problem L.I
Crystals can be reliably distinguished up to isometry only by isometry invariant I that takes the same value on all isometric sets. So I has no false negatives such that I(S) * I(Q) for different representations of isometric crystals S = Q. If descriptor has false negatives, we can make no reliable conclusions because different inputs as Fig. and (left) can reprsent isometric crystals.
Hence, non-invariants cannot used to justifiably compare crystals 77] It suffices to classify crystals up isometry f including reflections because, for any basis f1,...s : Un of R.. preserves orientation the determinant (f (V)) (u.)....,
for the images of Ui...., Un in the same basis.
The traditional approach to identifying a periodic crystal is to use its conventional or reduced cell [40 section 9.3]. This reduced cell has been known to be discontinuous under perturbations 44] even for 2D lattices when a motif M is one point. More formally, (26) section 1] and [67 Theorem 15) proved that a reduced cell cannot be continuously defined for all lattices. Discontinuity of many discrete invariants such as symmetry groups becomes clearer in Fig. (right) showing that even real-valued invariants struggle to continuously quantify the similarity between nearly identical sets. Under perturbations of points, a minimal cell U can double in size, while the density remains constant. Problem 1.1. Find a function on all periodic sets of unlabeled points in R" such that
L invariance: if any periodic point sets S = Q are isometric in TR. then T(S) = T(Q), so the invariant I has no false negatives;
I.1b) completeness: if I(S) I(Q) for any periodic point sets S, S.Q. then S Q are isometric, so the invariant has no false positives;
metric: a distance d between values of I satisfies all axioms; 1) d(1,, 1 I2) = 0 if and only if Tz, 2) symmetry J1, I2) = d(I,, I1), 3) triangle inequality d(I1, I2) + d(T2, 1a) 2 d[]1,) Ig);
I.Td) Lipschitz continuity: Q is obtained from a periodic set S C RD by shifting each point of S by most then d(I(S), < Cé for a constant C and any periodic point sets S, Q R";
I. computability: the invariant the metric d and verification of 1(S) I(Q) should be computed in near-linear time in the number of motif points of periodic sets for fixed dimension n.
inverse design: any periodic point set S C. 18 can be reconstructed from its invariant I(S).

---

Problem 1.1 is the ultimate data science challenge for all periodic crystals S whoss non-invarnn invariant input (a cell basis and motif) should be transformed into complete invariant [S),, which uniquely and unambiguously represents any S. Such a complete invariant can be considered as materials genome _2 or DNA-style code that also allows an explicit reconstruction for any periodic crystal.
For example, computer vision tries to identify humans or other objects such as road signs by using pixel-based images as input. Similar to other real objects, any periodic crystal can be given by (infinitely) many inputs. Problem [.] is an example of the data ambiguity challenge for almost any application that needs to uniquely identify real object. Biology has partially solved this challenge by a DNA code, though a living creature cannot be easily created from such a code. Condition (I.I) goes further and requires full reconstruction of periodic crystal S from its DNA-style code (S).
The proposed solution to Problem L.1 is the isometry invariant I called the Pointwise Distance Distribution PDD. Theorems 3.2
5.1 4.4 prove that PDD satisfies all conditions of Problem [.1 even (T.1) least for generic sets. More exactly, Theorem 4.4 shows that any periodie point set A M C IR" in a general position can be explicitly reconstructed from PDD and its lattice A.
4.3
The strength of PDD was experimentally checked for all 660K+ periodic crystals in the world's largest collection CSD (Cambridge Structural Database). Despite the CSD being curated to contain only real and distinct structures [30], the new invariants identified several pairs of duplicates. All the underlying publications are now being investigated for data integrity by five journals, see section 6
Problem 1.1 is stated in the hardest scenario when points are unordered and unlabeled because many real crystals have identical compositions. For example, diamond and graphite (whose 2-dimensional layer is graphene) both consist of pure carbon but have vastly different physical properties.
Conditions (d. for a continuous metric are stronger than a complete classification in (.I ab);
detecting an isometry gives discontinuous metric d(S, Q) = 1 (or another positive number) for all non-isometric 5 인 Q even if S, Q are near duplicates as Fig. _ (right). Continuity under perturbations is practically important because atoms vibrate, and any real measurement of a crystal produces slightly different coordinates of its unit cell and motif. Any simulation of periodic structures introduces floating point errors because of inevitable approximations by iterative optimization. Thousands of near-duplicates are routinely produced, though only a few structures are synthesized. Five real structures of 5679 predicted on a supercomputer over 12 weeks a typical example (29). This 'embarrassment of over-prediction'
[52] wastes time and resources to run simulations and then analyze results, often by visual inspection, because of a lack of invariants that solve Problem 1.1
Computability condition 1.1g avoids the trivial function T(S) = SS in Problem 1.1 Inverse design in (T.TD) allows one to replace the traditional blind sampling (of ambiguous cells and motifs leading to (near-) duplicates via optimization) with guided exploration of the crystal space parameterized by complete and reversible invariants. Section 2 shows that the state-of-the-arr tools remain stuck with conditions
Ib) while the new invariants satisfy the stronger practical requirements Ideff...
2 A review of the related state-of-the-art on comparing periodic point sets
Computational Geometry methods studied isometry questions related to Problem I.I for finite sets of points. The fastest algorithm to test an isometry between m-point sets in I2" runs in time Omm log m) by [3), which can be improved to O(m log m) in RR [41]. Significant results on matching bounded rigid shapes and registration of finite point sets were obtained in [56] 33 25
These methods focus on true/false answers without continuous and easily computable metrics 58 46
The classical isometry invariant of a finite set in any metric space is the distribution of pairwise distances who completeness in general position was proved in 2004 (12). The same paper described an infinite family of singular counter-examples
the full completeness, see the non-isometric 4-point sets in Fig. 3 (left). Persistent homology is a more recent isometry invariant of finite point sets, which continuous under perturbation but turned out to be weaker than previously anticipated Mémoli's 's seminal work on distributions of distances [45]. also known as shape
49 27 28
600
35 32 for bounded metric spaces is closest to the proposed Pointwise Distance Distributions (PDD) for periodic point sets. However, Problem 1.1 is not reducible to the finite sets by taking a cube or a ball of fixed (even very large) cut-off radius within a periodic point set. Indeed, one can easily find non-isometric subsets of the same periodic set in Fig. 4 (left).
distributions 48
9
38
3

---

r-8
-4
8+r
12
S(r)
1/ 110-
2
-8
r-6
2+r
10+r
110
-4
4
12 Q(
100
-8
r-6
n-4
2+r
4+r
10+r
12+f
Figure Left: point sets K = ((,,00), (#1, 1) and {(=2, 0), [1, can not be
distinguished by their six pairwise distances 22, 2,,,, v10 v10, Right: 1D periodic sets
S(r) = r0,r, 2 7,4] 82 and (r) {0, r, 7,4,4++1 82 for < S 1 have the same
Patterson function [51 p. 197, Fig. 2]. All these pairs are distinguished by PDD in section 3
Computer Vision methods focused on alignment or registration of point clouds in dimensions
3 [64, 53). The latest isometry invariant [17] is defined for scalar field R3 -> IRF which is more complicated than for finite cloud. Completeness up to isometry was resolved by the invariants [42] with continuous metrics that have polymomial time in the number of points for fixed dimension.
real periodic structure: input data (cell, motif)
9 points
points
8 points
points
ambiguous
discontinuous
points
4 points
non non-invariants
new invariant
pointwise distance distribution
PDD
points
points
points
points
MST
PDD
visualization by tree connecting close structures
Figure 4: Left: isometric periodic sets have many non-isometric finite subsets that cannot be used for comparison, discontinuity under perturbations is similar to Fig. 2 (right). Right: an ambiguous input is transformed into PDD to visualize any dataset as Minimum Spanning Tree (MST).
Crystallography and Materials Science developed many non-invariant descriptors by using quan- tities cell parameters and atomic coordinates in an ambiguous cell. The past approaches such as Symmetry Functions (8). rely on finite subsets with a fixed number of neighbors 18 or cut-off radius [37] see Fig. (left), without continuity in I.)). A reduction to finite-size subsets cannot guarantee a complete and continuous classification because, under tiny perturbations, minimal (by volume) cell can become larger than any subset of fixed size, see Fig. 2 (right).
SOAP (Smooth Overlapped Atomic Positions (7]), ACSF
and ACE (Atomic Cluster Expansion
Perturbations of atoms can change inter-atomic bonds depending on thresholds, which makes all graph-based descriptors [68 [0) discontinuous [6 65] and unreliable for detecting near-duplicates. If
threshold parameters change, then so (possibly, discontinuously) the invariants, see Fig.
crystal descriptors
density functions, complete isosets
generically complete, near-linear time: PDD
no invariant?
yes
fast? yes
no
useless non-invariants e.g. cell parameters, atomic coordinates no
isometry invariants
(non-constant useful)
strongest: distinguish almost all periodic sets
no false negatives
yes
continuous? yes
generically complete?
discontinuous invariants: reduced cell parameters, symmetries, space group
continuous invariants (more practically useful)
incomplete:
no SOAP, ACSF, powder diffraction patterns
quantify the similarity
Figure 5: A zoo of crystal descriptors. minimally useful (non-constant) invariant should have no false negatives. Most isometry invariants are discontinuous (70] or incomplete [50], hence fail conditions (I... The fastest and generically complete invariants PDD defined in this paper.

---

One of the oldest invariants of crystals is the X-ray diffraction pattern whose single crystal form is best for determining 3D structure of an experimental crystal [2]. Since not all materials can be grown as single erystals, powder X-ray diffraction patterns (PXRD) are more common. All periodic structures
with identical PXRDs are called homometric (44, see the periodic versions S(1) = (0, 1,3, 4) 8Z and Q(1) = {0,3, 5. + 8Z of the 4-point set T, K in Fig. 3 The more general sets 5(r),Q()) with the same Pair Distribution Function (PDF) will be distinguished by PDD in section 3
The CCDC's Mercury software compares periodic structures [8] by minimizing the Root Mean Square Deviation (RMSD) of atomic positions from up given number m (15 by default) of closest molecules in two structures. This comparison depends on many parameters (maximum number of matched molecules, thresholds for matched distances and angles), fails the triangle inequality in condition (I.I)) and is too slow for pairwise comparisons, see section 6 This RMSD tries to measure the maximum displacement of atoms only by using finite subsets, see the difficulties in Fig. (lef).
Any point p in I can be identified with the vector ş from the origin 0 to p. The Euclidean distance between p, g E R" is denoted by 1p 1p-gl, gl. which is the length of the difference vector p -g. For infinite periodic crystals, the theoretically better alternative is the bottleneck distance d Q) equal to the maximum Euclidean distance needed to perturb every point E S to its unique match in Q. Formally, (s, Q) = inf sup wp- - s()]l is minimized over all bijections g and maximized over g:S+Q PES
all points p E S for a fixed bijection. But dæ is not a metric on all periodic sets because d3 = +o0 for nearly identical lattices, and dæ is discontinuous under perturbations of lattice bases below.
Example 2.1 (infinite bottleneck). We show that S = Z and Q = (1 + 5) for any 5 >> 0 have dg(S (5, Q) = +o0. Assuming that ds(S,,) is finit, consider an interval [-N, N] C TR containing
2.N + 1 points of S. If there is a bijection g S - Q such that |p - g()) S dB for all points
p E S, the image of 2N + points S S N] under this bijection g should be within the interval
2(N+da) points, which is smaller than [- N - dR, N + dn)- The last interval contains only 1 + 148
1 + 2N when N+da < N dn < 8N. We gel a contradiction by choosing a large N >>> 5
1+6
If we consider only periodic point sets S. S,Qe C R" with the same density (or unit cells of the same size), the bottleneck distance dg (S, Q) becomes a well-defined wobbling distance [16).
Example 2.2 (discontinuity of the wobbling distance). Slightly perturb the basis 0). (0. 1) of the integer lattice Z? to the basis vectors (1, O), (e, 1) of the new lattice A. We prove that d Z²) 2 3 for any & > 0. Map R22 by 2 -r-translations to the unit square [0, 1]2 with identified opposite sides (a torus). Then the whole square lattice 2² is mapped to the single point represented by the corners of the square [0, 1J2. The perturbed lattice A maps to the sequence of points {KE (mod 1)) x {0, 1} in the horizontal edges. If (A, 222 =r < 3. then all above points should be covered by the closed disks of the radius r centered at the corners of [0, 1/2. For 0 <E <<> r, we can find k E Z so that ke is strictly between r, hence not covered by these disks, so d (A, Zz) 2
The discontinuity under perturbations is the major weakness of many past invariants including Voronoi diagrams, which should be matched via infinitely many rotations 477, space groups and other group-theoretic invariants [36]. The key example in Fig. 2 (right) shows that a continuous distance between nearly identical sets should be close to O, not exactly 0. These sets have different depending on manual thresholds (70).
symmetries and can be related only by ppcudo-symmette
For any k 2 1, the k-th density function 4/x (4) of periodic set S = A + M C R is the total volume of the regions within the unit cell U of A covered by exactly & balls B(p; t) with a radius 2 0 and centres at motif points p E M, divided by the cell volume Vol[U]. The density function (tx(t) was proved in (26 to be invariant under isometry, continuous under perturbations, complete for periodic sets satisfying certain conditions of general position in IR3, and computable in polynomial time in the motif size of S 6T]. Section 5 in [26]
Periodic Geometry develops invariants for Problem 1.1
gave the counter-example to completeness: the 1D sets S1s = X +-Y+112 and Q15 = X - y 152
for X = (0, 4. 9) and Y = (0, 1. 3) (39 section 4] have the same density functions for all k: 2 [ Example 10] but were distinguished in (77] Example 5b), [5] Example 6.3]. The latest advance 2
reduces Problem [.I to an isoset of isometry classes of a-clusters around points in motif at a certain radius which was motivated by the seminal work of Dolbilin [34]
23 The continuous metrics
on isosets 4. section 5] have only an approximate algorithm. Despite the progress in dimension 1
43 and lattices in dimension 2 [4 [5] and 3 42 14, Problem remained open in general.
5

---

The Pointwise Distance Distribution of a finite or periodic point set
Distances to neighbors considered in (67 Definition 5], though only their average proved to be invariant. Definition 3.1 introduces the matrix PDD whose ordered rows guarantee invariance in Theorem 3.2 and weights of rows that guarantee continuity under perturbations in Theorem 443
Definition 3.1 (Pointwise Distance Distribution PDD). For a motif of points M = {p1,.... Pm in unit cell U of lattice A. let S C. IN be a finite set coinciding with M
periodic set S = A M. For an integer 2 1, consider the m k matrix ((5: k), whose i-th row consists of the ordered distances du S .... I dus measured from ps its first nearest neighbors in the full set S. The rows of D(S; k) are lexicographically ordered as follows. A row (dıi,... dak) is smaller than (dgi...... ......dd.))) ) if a few first distances coincide: du -.ll...... da =d for I E {1,...,ke k 1} and the next (I + 1)-st distances satisfy di, d3,441- If rows are identical to each other, any such group is collapsed
one row with the weight w/m. For each row, put this weight in the first column. The final m x (k: 1)matri is the Pointwise Distance Distribution PDD(S; k).
The matrix 3) in Table has two pairs of identical rows, so the matrix PDD(T; consists of two rows weight & below. The matrix 3 in Table has only one pair of identical rows, so PDD(K;3) 3) has three rows weights 1/2 4.
4 3. Hence PDD (T;3) PDD(K; 3).
Table I: Each point in T, K C R² from Figure
3 has ordered distances to three other points.
points
neighb.1
neighb.2
neighb.3
K points
neighb.
neighb.2
neighb.3
(-2.0)
v2
V10
(-2,0)
V2
V2
(+2,0)
V
/10
4
(+2,0)
V10
V10
(-1,1)
V2
2
/10
(-1,-1)
V2
2
V10
(+1,
V
2
v10
(-1,+1)
V2
2
V10
1/4
v2
/2
4
1/2
v2
2
v10
PDD(T;3)
* PDD(K;3)
1/2
v2
2
v10
1/2
v2
v10
4
1/4
v10
v10
4
Theorem 3.2 (invariance PDD, all theorems are proved in the extended version [66]). For any
finite or periodic set S C IR,, PDD(S; k) from Definition 3.7is isometry invariant of S, k 2 1.
Table 2: Distances from each motif point of Sr)) and Q(r) to their closest neighbors in Fig.
Sr) points
distance to neighbor
distance to neighbor 2
distance to neighbor 3
P1
00 (2 (2+1) rr) r (2+rrr) =2
= r
0 =r
r 4 = -4- - (2+r) -0|=2+r
Ps +
(2+r) 2
(2+r) 2
= 4
4 (2+ 2-r
14 44-1= 44- -
44 0--0=4 =
Q(r) points
distance to neighbor 1
distance to neighbor 2
distance to neighbor 3
P1 = 2 +
0 0-(2+r)
r
(r- 8) =4
0 4|
(2+r)-+ 4 -r 4 (4++) =
(2+r) ()) = =
(2+r) 0 = 2rr
P3
44 =2
44-0
4 +r
(4+r) - =
=2
(4+ -8] - -
For the 1D periodic sets S(r) = {0, r, 2 + 8Z and Q(r)
+,,4++ 8Z with unit
cell [0, 8) in Fig. 3 Table 2
shows that S(r), Q(r) are not isometric for any parameter < S
1/4
2+r
4
6 r
- 8
PDD(S(r):8)
1/4
2
r
4
+ r
6
r 8
#
1/4
-
2
+ r
6
6
6+r
8
1/4
2-r
4 +
6+r
1/4
r
2
4
6 +
r
8-r
8
8
PDD(Q()):))
1/4
T
2
4 - r'
4 +++1
6
8-r
8
8
1/4
2 2-7
2
2
+
6 -
6
6+r
8 8
1/4
2+r
r
4
+
r
6-r
8 8

---

The averages of columns in (S:(S; form the vector AM k) of Average Minimum Distances (67] Definition 5]. Any lattice A R" has 1-point motif M {p). hence PDD(S; a single row of increasing distances from
all other points A {p}. Fig. 6 (right) shows honeycomb set whose motif consists of two points that have the same distances to all their neighbors, hence two rows of (SS; k) collapse to single vector. So PDD(S; k) (SD(S; k) in the cases above.
Example 3.3 (PDD is stronger than AMD). Since both sets (()). Qr)) in Fig. (right) have period the matrices PDD(S(r): k) and PDD(Q((): k) have distance in each row for columns and as shown above. All further distances are obtained from the first eight by adding a multiple of period
8. The vector AMD(S(()) k) of column averages for any l 2 8 is determined by AMD(S(r):
(1, 225,2.355 4.5, 5.5, 7, 8). Since the components of AMD(S(r)) k) do not depend the parameter 0 < r < 1, the sets Sr) are counter-example to the completeness of AMD, now distinguished by PDD(S(r); k) already for k Hence PDD(S; k) is strictly stronger than AMD(S; k).
PDD
PDD
2
PDD
55
2
3
2
3
k
1
1
Figure 6: The square lattice (left), hexagonal lattice (middle), and honeycomb periodic set (right) with minimum inter-point distance of 1 have PDD(S; k) with single row of increasing distances.
For periodic set S, the number & in PDD(S; k) can be considered degree of approximation (or a count of decimal places), not as a parameter that affects invariant values. If we increase k, we extract more distant geometric data from S by adding more columns to PDD(S; k) and keeping all previous distances. If some rows are identical in D(S: k 1) and become different in D(S; k). we recompute weights but not distances. The past tools [8 strongly depend on extra parameters.
Now compare PDD with the closest past invariant, the Pair Distribution Function (PDF). For periodic point set I with motif M, the exact ePDF consists ordered distances from all points p E M to all other points & S {p). So the infinite sequence ePDF(S) is obtained from PDD by combining distance from each pair p- sequences starting with (PDF(S(r)) = ePDF()(r)) = {r, - r, 2, + r, 4- r, 4, 4, 4 r....). This example shows that PDD for l is strictly stronger than ePDF
rows into one sequence and losing weights. Additionally, we keep only one pl. For any fixed 0 <r 1, the sets (r)), Q(r) have the same
isometry invariant.
For any lattice A C R", the vector ePDF(A; k) up to & distances concides with PDD(S; For the honeycomb set S in Fig. 6 (right), ePDF 2k) is obtained from PDD(S; k) by repeating every distance twice. If a periodic set is perturbed and a unit cell doubles as in Fig. 2 (right), then every distance in ePDF is replaced by couple of (near-) duplicate distances, eP (P discontinuously changes by including twice as many short distances and losing longer distances.
This typical discontinuity was roughly repaired by replacing every single distance d with its Gaussian distribution exp(-(z d /2o) with a parameter or > 0. Then a normalized sum of such 'blurred' distances 63 becomes the smooth Pair Distribution Function PDF(S; a)) Since algorithms can compare only finite vectors, this PDF(S; a)) is then uniformly sampled, which creates dependence on and sampling. PDD provides an easy alternative to this counter-intuitive PDF pipeline (discrete sequence) + {smooth function] {discrete sequence). whose continuity wasn't 't formally proved.
7

---

4 Continuity and generic completeness of Pointwise Distance Distributions
Continuity of PDDD(S; k) under perturbations of S in the bottleneck distance dg will be measured by the Earth Mover's Distance [55], which can be applied to any weighted distributions of different sizes.
Definition 4.1 is for any vector I(S) = ([w1 (S)).. [wm(s)) Rm(s) (5)]) of pointwise
m(())
invariants of a set 5 with weights ws( E (0, 1] satisfying the normalization
Σ ws(S) 1.
Later we consider only the case when the weighted vector [ws, Ra] is the i-th row of PDD(S; (S; k). Then (S) is the number of rows in PDD(S; (S; k). Each row (S) should have a size independent of S, for example a number k of neighbors in PDD(S;k) k). For any vectors R. = (Ta1..... Tss) and R (ry1.... T33)) of length k we use the -iistantee RR Rylo= max Ira rytles.
(=l...,
Definition 4.1 (EMD). Let finite or periodic sets S, Q C TR" have weighted vectors T(S), [(Q) as discussed above. A flow from T(S) to T(Q) is an m(S) x m(Q) matrix whose element fiz E (0, 1] represents partial flow from (S) to (Q). The Earth Mover's Distance is the minimum cost
m)
mlQ)
miQ)
EMD(I()),X(1)
Σ
Σ
fu (S) - (Q) fig E [0,1] subject to
Σ
fig S (S)
1
j=1
j=1
m(S)
m(5) m())
for I....,,,(... m()), Σ fag S w (Q) for j 1.......
...... m(Q). .,.....((.)), and Σ
fg = 1.
m(Q)
The first condition Σ fiz S (S) means that not more than the weight (S) of the component
j=1
R((S) 'flows' into all components (Q) via 'flows' Jig. = 1,..., m(Q). Similarly, the second m(S)
condition
Σ Jag =Q) means that all 'flows' Si from (S) for
...... 'flow"
m(S) m())
into (2) up to the maximum weight wy (Q). The last condition Σ Σ fu forces to
'flow' all rows R₂ (S) to all rows (Q). The EMD satisfies all metric axioms 55 appendix], needs
O(m" log m) time for distributions of a maximum size m and is approximated in ((m) time [59 57).
Theorem 4.2 (lower bound of EMD). For finite or periodic point sets S.Q R" and k 2 1, the
distances satisfy EMD(PDD(S: S; PDD(Q k)) 2 ||AMD(S; k) - (
Theorem 4.3 uses the bottleneck distance d5(S Q)
inf sup lp g(p)l and the packing radius g:S 9: PES
r(S), which is the minimum half- half-distance between any points of S. Equivalently, r(S) is the maximum radius r to have disjoint open balls of radius centered at all points of S.
Theorem 4.3 (continuity of PDD). For any k 2 1, if finite or periodic sets S, Q C R" satisfy dø(S, Q) < r(S), then EMD(PDD(S;)), PDD(Q < 2d
Continuity Theorem 4.3 means that any small perturbation of atomic positions in the bottleneck distance d leads to a small change of the Pointwise Distance Distribution in the Earth Mover's Distance. Theorem 4.3 extends the following fact for 2-point sets (k- = 1). If we perturb two points by most e, the distance between them changes by at most 2e.
For any set C 1" of m points with distinct inter- point distances, completeness of PDD(S; m - 1) follows from [67 Theorem 16). Following the earlier work (26) section 5. the extended version [66] defines a distance-generic set that can approximate any periodic point set S = A + M C I.". The number of points in unit cell U is isometry invariant because any isometry maps U to another cell with the same number m of points. Theorem 4.4 assumes that a lattice A is given and reconstructs periodic point set S A + M in any dimension n 2 2. However, in low dimensions n = 2,3, any lattice A can be reconstructed from its isometry invariants described in 19
Theorem 4.4 (generic completeness of PDD). Let = A + M C R be a distance-generic periodic set with m points in motif M. Let R(A) be the smallest radius such that all closed balls with centers
p E A cover R.. Let 2(()) be smaller than all distances in the last column of PDD(SS k) for a big enough k. The set S can be uniquely reconstructed up to isometry from A. m. and PDD(S; k).
8

---

Polynomial time algorithms and experimental comparisons of PDD
The algorithm for PDD in Theorem 5.1 found several pairs of unexpected duplicates, which were missed by all past tools, through 200B+ pairwise comparisons of 660K+ real periodic crystals over couple of days on AMD Ryzen 5 5600X (6-core) @4.60Ghz, 32GB DDR4 RAM 3600 Mhz.
The main input is periodic point set S C R" given by unit cell U and motif of points in U. The key parameters of PDD(S: S: k) are m and k. The complexity in Theorem 5.1 is near-lineat linear in both & and m for fixed dimension and exactly lo + columns. The first column contains the weights of rows, which sum to and are proportional to the number of appearances of the row before collapsing,
The output PDD(S; k) is matrix with at most rows
the detailed code in [66]
Theorem 5.1 (PDD complexity). Let a periodic S IR have m. points in a unit cell U. For fixed dimension n, PDD(S; k) is computed in near-linear time (Kkm(5))" Vn log(m) log" (k)), where Vn is the unit ball volume in R", d and
d are the diameter and skewness of U.
VVoljj]
The near-linear time complexity Theorem goes far beyond the state-of-the-art. Section reviewed that all past tools are based ambiguous con-invariant data discontinuous invariants that miss (near- Jdupliccas,,,, or are too slow for pairwise comparisons of millions of crystal structures. The recent continuous invariants with theoretical (not exactly computable) metrics [26 section 6] and [ section 8] require cubic algorithms, which turned out to be unrealistic for large crystal datasets.
The Cambridge Crystallographic Data Centre (CCDC) has curated the world's largest collection of real solid crystalline materials since the 1960s, now having nearly 1.2M known structures. New crystalline material is deposited in the CSD only after peer-reviewed publication. The CCDC should ideally check that a new structure is genuine and not a duplicate of an earlier one because their data is trusted by all pharmaceutical giants developing new drugs in crystalline form. However, the CSD basically huge list of Crystallographic Information Files representing crystals by unit cells and motifs of points in coordinates of cell basis with limited search and comparison capabilities.
The papers 29 69 reported five experimental T2 crystals (based on the same molecule T2) that successfully synthesized after predicting 5679 crystals through 12-week simulations supercomputer. All initial 2M+ randomly sampled crystals were iteratively optimized to the 'most stable' approximations of local energy minima. This is typical 'embarrassment of over-prediction' when many (near-)duplicates are found around the same local minimum but remain undetected.
One striking example is the pair of crystals 14 and 15 in Fig. reported in 29 which have very different unit cells but are nearly identical. When this pair was compared by another free software Platon (62), bug discovered, which required few months to be fixed. Such bugs will keep emerging because the discontinuity of past invariants and metrics was not addressed as in Problem I.]
Figure 7: Crystals 14, 15 based on the T2 molecule have very different Crystallographic Information Files (with different motifs in unit cells of distinct shapes) but are nearly identical up isometry.
For example, rough sampling of the density functions (1) from 26 of 5679 crystals for up to k took more than four days on a comparable machine. This experiment detected the T2-5 crystal that was accidentally not deposited in the CSD because of visual confusion with another structure.

---

6 Conclusions and discussions of current limitations and duplicate detection
The popular packing similarity [8] algorithm COMPACK is available in the CCDC's Mercury software. The 4950 comparisons of the 100 lowest energy crystals close to T2-ô in density by packing similarity took hours 53 min, 2.8 seconds. Extrapolating this time for comparing any new structure with the whole CSD gives 38 days, hence more than 34 thousand years for pairwise comparisons of all 660K+ periodic crystals in the CSD. A typical comparison by AMD and PDD runs in nanoseconds and milliseconds, respectively. so comparing 100 crystals pairwise takes less than one minute.
Table 3: Most comparisons of 100 lowest energy crystals close to the T2-6 by packing similarity matched small numbers of molecules for the default maximum 15, which means failure to match.
[8
molecules
2
6
7
8
9
10
11
12
13
14
15
comparisons 2784
1150
773
69
85
21
31
6
2
7
12
3
1
0
6
Most importantly, more than half of all comparisons in Table matched only one molecule from two crystals. Since all crystals consist of the same rigid molecule T2, this ouiput means a complete failure: one common molecule, no other conclusions. Since the CCDC deposits hundreds of new structures daily, the shortcut approach is to compare the chemical composition of atoms. But even water (H2O) has least 15 forms of ice crystals, while other compositions have many more polymorphic forms in the CSD. This comparison by composition can miss duplicates where one atom is incorrect.
The unreliability of past comparisons fueled the growing crisis of fake crystallographic data [T] leading to the investigation of nearly 1000 structures in the CSD. The new invariants PDD detected five pairs of unexpected duplicates ('needles in the haystack') in the Cambridge Structural Database (CSD). First, the simpler invariants AMD(S) S; 100) were computed for all 660K+ periodic structures in the CSD, without disorder and with full geometric data. More than 200 billion pairwise comparisons of AMD(S; 100) vectors revealed 6371 pairs S, Q with AAM S: 100) AM;(Q; 100) sc S 0.01. As an AMD is simpler and faster compare, up the order of 10-7 seconds per comparison, this step took around 8 hours. This example and Theorem 4.2 show AMD is good filter.
Second, computing the -ased EMD between the pairs above detected 182 pairs with EMD < 0.01. Most of these pairs were expected and were the same crystal, or different aliases for the same database entry. The five pairs reported in 66 section 7] were unexpected because the underlying periodic sets of points at atomic centres were truly isometric (to the last decimal place) but one atom had different chemical elements in two crystals. The crystals with the CSD codes HIFCAB and JEPLIA are literally isometric but one Cadmium is replaced by Manganese at the same position.
All past tools taking into account atomic types see these geometric duplicates as different. The CCDC agreed that such a coincidence is physically impossible because another atom should have slightly different distances to neighbors detected by PDD. Hence at least one of the structures in each pair cannot be correct. The five journals are investigating the data integrity of the underlying publications. Detecting duplicates might have some negative impact on past publications that could be retracted
In conclusion, Theorems 3.2
fulfilled almost all conditions of Problem I.] while all past tools remained discontinuous or too slow for billions of comparisons. The only limitation is hypothetical existence of singular sets S 전 Q with PDD(S; k) = PDD(Q; k) for all A 2 1. We found no such examples among all real materials in the CSD. Even if such a singular pair of S, Q emerges, they can be distinguished by the slower and provably complete invariant isoset
Hence PDD and complete isoset can be considered as a DNA-style code or materials genome uniquely and continuously identifying any periodic crystal. More importantly, the experiments confirmed the Crystal Isometry Principle [67] section 7]: the map {periodic crystals) injective (doesn't 't lose information) modulo isometry. Hence all existing and not yet undiscovered crystals live in the common Crystal Isometry Space (CRISP) of isometry classes of periodic point sets continuously parameterized by complete isometry invariants. This CRISP can be considered a continuous universe containing all known crystals as visible stars in the universe, which requires mapping and exploration. Since inter-atomic distances cannot be arbitrary, the next step is to describe subspaces of CRISP that represent realistic but still unknown materials. This first map of this space for 2D lattices appeared in [15). We thank all reviewers for their valuable time and suggestions.
{periodic point sets} is
10

---

References
[1] Olga Anosova and Vitaliy Kurlin. arXiv:2103.02749.
Introduction to periodic geometry and topology.
[2] Olga Anosova and Vitaliy Kurlin. An isometry classification of periodic point sets. Lecuure Notes in Computer Science (Proceedings of DGMM), volume 12708, pages 229-241, 2021.
[3] Olga Anosova and Vitaliy Kurlin. Density functions of periodic sequences. In Lecture Notes in Computer Science (Proceedings of DGMM), volume 13493, pages 395-408, 2022.
[4] Olga Anosova and Vitaliy Kurlin. Recognition of near-duplicate periodic patterns by polynomial- time algorithms for a fixed dimension. arxiv:2205. 15298, 2022.
[5] Olga Anosova and Vitaliy Kurlin. Density functions of periodic sequences of continuous events. drxiv:2301.051377 2023.
[6] John M Ball and Stephen J Bedford. Discontinuous order parameters in liquid crystal theories. Molecular Crystals and Liquid Crystals, 612(1):1-23, 2015.
[7] Albert P Bartók, Risi Kondor, and Gábor Csányi. On representing chemical environments. Physical Review 87(18): 2013.
[8] Jörg Behler. Atom-centered symmetry functions for constructing high-dimensional neural network potentials. The Journal of chemical physics, 134(7):074106, 2011.
[9] Serge Belongie, Jitendra Malik, and Jan Puzicha. Shape matching and object recognition using shape contexts. Transactions PAMI, 24(4):509-55)) 52222, 2002.
[10] David Bimler. Better living through coordination chemistry: A descriptive study of prolific papermill that combines crystallography and medicine. 2022.
[11] M. Bouniaev and N. Dolbilin. Regular and multi-regular t-bonded systems. J. Information Processing, 25:735-740, 2017.
[12] M. Boutin and G. Kemper. On reconstructing n-point configurations from the distribution of distances or areas. Adv. Appl. Math., 32(4):709-735, 2004.
[13] Peter Brass and Christian Knauer. Testing the congruence of d-dimensional point sets. In Proceedings of SoCG, pages 310-314, 2000.
[14] Matthew Bright, Andrew Cooper, and Vitaliy Kurlin. Welcome to a continuous world of 3-dimensional lattices. arxiv:2109.11338 2021.
[15] Matthew J Bright, Andrew Cooper, and Vitaliy Kurlin. Geographic-style maps for dimensional lattices. Acta Crystallogruphica Section A, 79(1), 2023.
[16] Hans-Georg Carstens, Walter A Deuber, Wolfgang Thumser, and Blke Koppenrade. Geometrical bijections in discrete lattices. Combinatorics, Probability and Computing, 8:109-129, 1999.
[17] Laurent Chauvin, William Wells III, and Matthew Toews. Registering image volumes using 3D SIFT and discrete SP-symmetry. arXiv:2205. 1556, 2022.
[18] J.
Chisholm and S. Motherwell, Compack: program for identifying crystal structure similarity using distances. J. Applied Cryst., 38:228-231, 2005.
[19] J Conway and N Sloane. Low-dimensional lattices. VI. Voronoi reduction of diree-dimensional lattices. Proceedings Royal Society A, 436(1896):55-688, 1992.
[20] Kishalay et al Das. CrysXPP: An explainable property predictor for crystalline materials. npj Computational Materials, 8(1):1-11, 2022.
[21] William IF David, Kenneth Shankland, Ch Baerlocher, LB McCusker, et al. Structure determi- nation from powder diffraction data, volume 13. 2002.

---

[22] de Pablo et L. New frontiers for the materials genome initiative, upj Computational Materials, 5(1):1-23, 2019.
[23] N Dolbilin and M Bouniaev. Regular t-bonded systems in R European Journal of Combina- lorics, 80:89-101, 2019.
[24] Ralf Drautz. Atomic cluster expansion for accurate and transferable interatomic potentials. Physical Review B, (9))00144, 2019.
[25] Dym and Kovalsky. Linearly converging quasi branch and bound algorithms for global rigid registration. In Proceedings ICCV, pages 1628-1636, 2019.
[26] Edelsbrunner, T Heiss, V Kurlin, P Smith, and M Wintraecken. The density fingerprint of periodic point set. In Proceedings of SoCG, pages 32:1-32: 2021.
[27] Y. Elkin and V. Kurlin. The mergegram of dendrogram and its stability. In Proceedings of MFCS (Mathematical Foundations of Computer Science), 2020.
[28] Y. Elkin and V. Kurlin. Isometry invariant shape recognition of projectively perturbed point clouds by the mergegram extending Od persistence. Mathematics, 9(17), 2021.
[29] A Pulido et al. Functional materials discovery using energy-Structure maps. Nature, 543:657- 664, 2017.
[30] Groom et al. The Cambridge Structural Database. Acta Cryst B, 7222:171 2016.
[31] H Maron et al. Point registration via efficient convex relaxation. ACM Trans. Graphics, 35(4):1-12, 2016.
[32] H Pottmann et al. Integral invariants for robust geometry processing. Comp. Aided Geom. Design, 26(1):37-60, 2009.
[33] J Yang et al. Go-ICP: A globally optimal solution to 3D ICP point-set registration. Transactions PAMI, 38:2241-2254, 2015.
[34] Dolbilin et al. Multiregular point systems. Discrete Computational Geometry, 20(4):477- 498, 1998.
[35] Manay et al. Integral invariants for shape matching. Trans. PAMI, 28:1602-1111, 2006.
[36] Giuseppe Fadda and Giovanni Zanzotto. On the arithmetic classification of crystal structures. Acta Cryst. A: Foundations, 57(5):492-506, 2001.
[37] Flor, Orobengoa, E Tasci, Perez-Mato, and M Aroyo. Comparison of structures applying the tools available at the bilbao crystallographic server. J. Appl. Cryst., 49:653-66 2016.
[38] Cosmin Grigorescu and Nicolai Petkov. Distance sets for shape filters and shape recognition. IEEE transactions image processing, 12110):1274-1288 2003.
[39] Grünbaum and Moore. The of higher-order invariants in the determination of generalized Patterson cyclotomic sets. Acta Cryst A. 51:310-323, 1995.
[40] T Hahn, U Shmueli, and J Arthur. Internatationa tables for crystallography, volume 1. 1983.
[41] Heuna Kim and Günter Rote. Congruence testing of point sets in dimensions. 1003.066,,,, 2016.
[42] Kurlin. A complete isometry classification of 3D lattices. arxiv:2201.10543 2022.
[43] Vitaliy Kurlin. Exactly computable and continuous metrics on isometry classes of finite and
1 1-periodic sequences. arxi::220500436. 2022.
[44] Vitaliy Kurlin. Mathematics 2-dimensional lattices. Foundations of Computational Mathematics, 2022.
12

---

[45] Facundo Mémoli. Gromov-Wassersee Wasserstein distances and the metric approach to object matching. Foundations of Comp. Mathematics, 1 1(4):417-487 2011.
[46] Facundo Mémoli, Zane Smith, and Zhengchao Wan. The Gromov-Hausdorff distance between ultrametric spaces: its structure and computation. arXiv:2110.03136, 2021.
[47] Marco M Mosca and Vitaliy Kurlin. Voronoi-based similarity distances between arbitrary crystal lattices. Crystal Research and Technology, 1505): 2020.
[48] Robert et al Osada. Shape distributions. Transactions on Graphics, 21:807-8255 03277 2002.
[49] Steve Oudot and Elchanan Solomon. Inverse problems in topological persistence. In Topological Data Analysis, pages 405-433 Springer, 2020.
[50] B Parsaeifard and S Goedecker. Manifolds of quasi-constant SOAP and ACSF fingerprints and the resulting failure to machine learn four-body interactions. J Chem. Phys., 156:034302, 2022.
[51] Patterson. Ambiguities in the X-ray analysis of structures. Phys. Rev, 63:195-201, 1944.
[52] S Price. Is zeroth order crystal structure prediction (csp_0) coming to maturity? what should aim for in an ideal crystal structure prediction code? Faraday discussions, 211:9-30, 2018.
[53] Blaine Rister, Mark A Horowitz, and Daniel L Rubin. Volumetric image registration from invariant keypoints. Transactions on Image Processing, 26(10):490044910, 2017.
[54] Joseph Rosenblatt and Paul D Seymour: The structure of homometric sets. SIAM Journal on Algebraic Discrete Methods, 3(3):343-350, 1982.
[55] Y. Rubner, C. Tomasi, and L. Guibas. The earth mover's distance as metric for image retrieval. Intern. Journal of Computer Vision, 40(2):99-121, 2000.
[56] Mauro R Ruggeri and Dietmar Saupe. Isometry-invarian matching of point set surfaces. In Eurographics Workshop on 3D Object Retrieval, pages 17-24. Citeseer, 2008.
[57] Ryoma Sato, Marco Cuturi, Makoto Yamada, and Hisashi Kashima. Fast and robust comparison of probability measures in heterogeneous spaces. arXiv:2002.01615. 2020.
[58] Felix Schmiedl. Computational aspects of the Gromov-Hausdorff distance and its application non-rigid shape matching. Discrete and Computational Geometry, 57(4))855-88) 880, 2017.
[59] Shirdhonkar and D Jacobs. Approximate earth mover's distance in linear time. In Conference on Computer Vision and Pattern Recognition, pages 1-8, 2008.
[60] Philip Smith and Vitaliy Kurlin. Families of point sets with identical ID persistence,. arxiv:2202.00577, 2022.
[61] Philip Smith and Vitaliy Kurlin. A practical algorithm for degree-k Voronoi domains of 3D periodic point sets. In Lecture Notes Comp. Science (ISVC), volume 13599, 2022.
[62] A Spek. Single-crystal structure validation with PLATON. J Appl. Cryst., 36:7-13, 2003.
[63] Maxwell W Terban and Simon JL Billinge. Structural analysis of molecular materials using the pair distribution function. Chemical Reviews, 122: 12722, 2022.
[64] Matthew Toews and William M Wells III. Efficient and robust model-to --oaaa alignment using 3D scale-invariant features. Medical image analysis, 17(3):271-222, 2013.
[65] Anne Mai Wassermann, Mathias Wawer, and Jurgen Bajorath. Activity landscape representa- tions for structure-activity relationship analysis. J Med. Chemistry, 53(23):8220 8223, 2010.
[66] D Widdowson and V Kurlin. Pointwise distance distributions. wwxiv22108.04790 2021.
[67] Daniel Widdowson, Marco M Mosca, Angeles Pulido, Andrew I Cooper, and Vitaliy Kurlin. Average minimum distances of periodic point sets foundational invariants for mapping all periodic crystals. MATCH Comm. in Math. and in Computer Chemistry, 87:529-559, 2022.
13

---

[68] Tian Xie and Jeffrey C Grossman. Crystal graph convolutional neural networks for an accurate and interpretable prediction of material properties. Phys. Rev. Letters, 120:145301, 2018.
[69] Q Zhu, J Johal, D Widdowson, Z. Pang, B Li, C Kane, V Kurlin, G Day. M Little, and A Cooper. Analogy powered by prediction and structural invariants: Computationally discovery of a
mesoporous hydrogen-bonded organic cage crystal. J Amer. Chem. Soc., 144:9893-9901, 2022.
[70] Peter Zwart, Ralf Deesss-Kunss,eee,, Andrey Lebedev, Garib Murshudov, and Paul Adams. Surprises and pitfalls arising from (pseudo) symmetry. Acta Cryst. D, 64:99-107, 2008.
Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the pa- per's contributions and scope? [Yes] The main theoretical results are stated Theo-
rems 3.2 4.3 5.1 4.4 and are proved in the supplementary materials [66]. The key
experiments are described in sections with more details in the full version.
(b) Did you describe the limitations of your work? [Yes] Yes, discussed in section
(c) Did you discuss any potential negative societal impacts of your work? [Yes] Potential fraud and possible retractions of several papers are discussed in section 6
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] In particular, the paper properly anonymized.
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] Yes, all formal theorems are completely stated. For Theorem 4.4 the definition of a distance-generic
periodic point set couldn't 't fit the page limit and is in the supplementary materials (66).
(b) Did you include complete proofs of all theoretical results? [Yes] Yes, in the supple- mentary materials or the extended version of this conference paper [66].
3. If you experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main exper- imental results (either in the supplemental material or as a URL)? [Yes] Yes, in the supplementary materials [66].
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] The PDD algorithm used one parameter k = 100 (the number of atomic neighbors) in addition to typical input (Crystallographic Information File). Increasing adds more distances to PDD(S; k) without changing the previous values. (c) Did you report error bars (e.g., with respect the random seed after running experi- ments multiple times)? [Yes] Yes, the threshold of 10-12 meters (atomic scale) was used identify and further investigate (near- dduplccdu structures, however all exact duplicates were reported at distance exactly O.
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [Yes] Yes, the specifications of the modest desktop computer appear at the beginning of section 5
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes] Yes, the Cambridge Structural Database (30] and the database of T2 crystals reported in (2
(b) Did you mention the license of the assets? [Yes] Individual structures can be freely downloaded from the Cambridge Structural Database (CSD) by their 6-letter codes
given the paper and supplementary materials.
(c) Did you include any assets either in the supplemental material as a URL? [Yes] For convenience, the supplementary materials include the pairs of duplicate structures and other important crystals mentioned in the paper.
(d) Did you discuss whether and how consent was obtained from people whose data you're 're esing/curating? [Yes] All used data is freely available.
(c) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] All data is non-personal.
14


Reviews:
[{"review_id": "zw8BMb4LQ-", "reviewer": "Reviewer_wsDJ", "summary": "This paper focuses on the problem of geometric isomorphism for periodic structures of materials. They propose a new crystal descriptor without false negatives, which means if two periodic structures are the same, the output of this PDD descriptor will be the same. This PDD descriptor has a near-linear time complexity. They achieve this by formulating a PDD matrix using k-nearest pairwise distances for all atoms in a unit cell.", "questions": "My main concerns are the weaknesses 2 and 3 mentioned above, could you please give some clarifications about them? ", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n* This proposed descriptor has near-linear time-complexity without false negative when only the positions of points considered.\n* The illustration of different unit cell representations for the same material is insightful and easy to understand.\n* This proposed method can deal with a larger unit cell representation as shown in Fig.2 in the main paper.\n\n## Weaknesses\n* Originality: The idea of using k-nearest pairwise distances is not new. CGCNN proposed by Xie et al. actually uses k-nearest pairwise distances to form the crystal graph representation to make the graph input invariant to different unit cell representations. \n* The false positive, which is defined in 1.1b, can not be satisfied when the k is given (large enough k can not be achieved during usage of this PDD descriptor). The increase of k will  increase the time-complexity at least linearly. \n* The false negative, which is defined in 1.1a, can not be satisfied when you consider atomic numbers of points. Two different materials with the same geometric structure and a bit different atomic numbers will result in the same output of PDD descriptor. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "vYt_GJrx1o", "reviewer": "Reviewer_FXDP", "summary": "This paper provides a method based on the Point-wise Distance Distribution PDD(S; k) to describe/characterize periodic crystal arrangements, i.e. atomic/molecular configurations. \nThis can be used to efficiently identify instances of similar periodic crystal patterns in large datasets.\nEssentially, to find a function I on all periodic sets of unlabeled points in Rn such that invariance, completeness, metricity, continuity, computability and inverse design criteria hold.\nProofs are shown.\nVarious experiments demnostrate the system on atomic crystal configurations from the Cambridge Crystallographic Data Centre (CCDC).", "questions": "Is this method relevant to general volumetric AI, where geometric structure is generally not be periodic?\n\nOther approaches such as the Hough transform can be used to identify and classify non-periodic 3D atom-like structures with high accuracy, which invariance to scale, translation, rotation. It seems such a method work well in predicting the periodic structures shown here, also in a manner invariant to scaling. Is this the case?\nChauvin, L., et al. 2022. Registering Image Volumes using 3D SIFT and Discrete SP-Symmetry. arXiv preprint arXiv:2205.15456.\n\n\n", "limitations": "Limited to periodic atomic crystal structures.\n\nAlternative methods based on non-periodic atomic 3D image structures may outperform this method.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: the method apparently works well for indexing crystals, proofs are shown as to the method here, based on distance distributions.\n\nWeakness: predicting periodic crystal structures is a niche domain, it is unclear how this is relevant to general DL.\n   Lack of citation to directly relevant research.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "rZme34btJUp", "reviewer": "Reviewer_TtwX", "summary": "The authors provide a periodic-invariant feature called Pointwise Distance Distributions (PDD). The feature is constructed by distances to the first k nearest neighbors of every point in a periodic set with near-linear complexity. The authors also prove that this feature can remove ambiguity due to different representations of a crystal cell, is invariant that different basis of the same material won’t change output, and is continuous that small perturbation won’t largely change the output. The authors use this feature to remove duplicate materials and perform their experiments on the Cambridge Structural Database. The effectiveness of the method is shown by less time and fewer failures.", "questions": "Figure 3 Left: Why can’t we distinguish between these two graphs? The distance matrices of them are different.", "limitations": "The method is not novel enough. I think other methods such as the radius graph or the combination of radius graph and k-NN graph also suffice to distinguish materials.\n\nFor experiments, I recommend using PDD for other material databases to remove variance of your experiments; and also using PDD as features of other important tasks such as material energy prediction.\n", "rating": 5, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\nThe authors give many examples to show problems of periodic representation and the proof of continuity and completeness of PDD is well-stated.\n\nWeakness:\nUsing the k-nearest distances to build the features is not novel enough. It’s easy to build a crystal graph with k-nearest neighbors. The experiment is only done in one database and the task is not significant enough. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept