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Molecular set transformer is a deep learning architecture for scoring molecular pairs found in co-crystals, whilst tackling the class imbalance problem observed on datasets that include only successful synthetic attempts.

The fundamental model of any solid crystalline material (crystal) at the atomic scale is a periodic point set. The strongest natural equivalence of crystals is rigid motion or isometry that preserves all inter-atomic distances. Past comparisons of periodic structures often used manual thresholds, symmetry groups and reduced cells, which are discontinuous under perturbations or thermal vibrations of atoms. This work defines the infinite sequence of continuous isometry invariants (Average Minimum Distances) to progressively capture distances between neighbors. The asymptotic behaviour of the new invariants is theoretically proved in all dimensions for a wide class of sets including non-periodic. The proposed near linear time algorithm identified all different crystals in the world's largest Cambridge Structural Database within a few hours on a modest desktop. The ultra fast speed and proved continuity provide rigorous foundations to continuously parameterise the space of all periodic crystals as a high-dimensional extension of Mendeleev's table of elements.

Mesoporous molecular crystals have potential applications in separation and catalysis, but they are rare and hard to design because many weak interactions compete during crystallization, and most molecules have an energetic preference for close packing. Here, we combine crystal structure prediction (CSP) with structural invariants to continuously qualify the similarity between predicted crystal structures for related molecules. This allows isomorphous substitution strategies, which can be unreliable for molecular crystals, to be augmented by a priori prediction, thus leveraging the power of both approaches. We used this combined approach to discover a rare example of a low-density (0.54 g cm -3) mesoporous hydrogen-bonded framework (HOF), 3D-CageHOF-1. This structure comprises an organic cage (Cage-3-NH 2) that was predicted to form kinetically trapped, low-density polymorphs via CSP. Pointwise distance distribution structural invariants revealed five predicted forms of Cage-3-NH 2that are analogous to experimentally realized porous crystals of a chemically different but geometrically similar molecule, T2. More broadly, this approach overcomes the difficulties in comparing predicted molecular crystals with varying lattice parameters, thus allowing for the systematic comparison of energy-structure landscapes for chemically dissimilar molecules.

The implementation of machine learning models has brought major changes in the decision-making process for materials design. One matter of concern for the data-driven approaches is the lack of negative data from unsuccessful synthetic attempts, which might generate inherently imbalanced datasets. We propose the application of the one-class classification methodology as an effective tool for tackling these limitations on the materials design problems. This is a concept of learning based only on a well-defined class without counter examples. An extensive study on the different one-class classification algorithms is performed until the most appropriate workflow is identified for guiding the discovery of emerging materials belonging to a relatively small class, that being the weakly bound polyaromatic hydrocarbon co-crystals. The two-step approach presented in this study first trains the model using all the known molecular combinations that form this class of co-crystals extracted from the Cambridge Structural Database (1722 molecular combinations), followed by scoring possible yet unknown pairs from the ZINC15 database (21 736 possible molecular combinations). Focusing on the highest-ranking pairs predicted to have higher probability of forming co-crystals, materials discovery can be accelerated by reducing the vast molecular space and directing the synthetic efforts of chemists. Further on, using interpretability techniques a more detailed understanding of the molecular properties causing co-crystallization is sought after. The applicability of the current methodology is demonstrated with the discovery of two novel co-crystals, namely pyrene-6H-benzo[c]chromen-6-one (1) and pyrene-9,10-dicyanoanthracene (2)., Machine learning using one class classification on a database of existing co-crystals enables the identification of co-formers which are likely to form stable co-crystals, resulting in the synthesis of two co-crystals of polyaromatic hydrocarbons.

A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard reductions remained discontinuous under perturbations modelling crystal vibrations. This paper completes a continuous classification of 2-dimensional lattices up to Euclidean isometry (or congruence), rigid motion (without reflections), and similarity (with uniform scaling). The new homogeneous invariants allow easily computable metrics on lattices considered up to the equivalences above. The metrics up to rigid motion are especially non-trivial and settle all remaining questions on (dis)continuity of lattice bases. These metrics lead to real-valued chiral distances that continuously measure a lattice deviation from a higher-symmetry neighbour., 51 pages, 20 figures. This version has the full appendix. The new continuous analogues of binary chirality were renamed as G-chiral distances to avoid potential confusion with the classical concept. The latest version is at http://kurlin.org/projects/periodic-geometry-topology/lattices2Dmaths.pdf

Packings of regular convex polygons (n-gons) that are sufficiently dense have been studied extensively in the context of modeling physical and biological systems as well as discrete and computational geometry. Formerly most of the results were on densest lattice or double-lattice configurations. Here we extend these results to all 2-dimensional crystallographic symmetry groups (plane groups) by restricting the configuration space of the general packing problem of congruent copies of a compact subset of the 2-dimensional Euclidean space to particular isomorphism classes of the discrete group of isometries. We formulate the plane group packing problem as a nonlinear constrained optimization problem. By the means of the Entropic Trust Region Packing Algorithm that solves this problem approximately, we examine some known as well as unknown densest packings of n-gons in all 17 plane groups and state conjectures about common symmetries of the densest plane group packings of all n-gons.

Rigid shapes should be naturally compared up to rigid motion or isometry, which preserves all inter-point distances. The same rigid shape can be often represented by noisy point clouds of different sizes. Hence the isometry shape recognition problem requires methods that are independent of a cloud size. This paper studies stable-under-noise isometry invariants for the recognition problem stated in the harder form when given clouds can be related by affine or projective transformations. The first contribution is the stability proof for the invariant mergegram, which completely determines a single-linkage dendrogram in general position. The second contribution is the experimental demonstration that the mergegram outperforms other invariants in recognizing isometry classes of point clouds extracted from perturbed shapes in images., arXiv admin note: substantial text overlap with arXiv:2007.11278

Author(s): Rutz, JJ; Shields, CA; Lora, JM; Payne, AE; Guan, B; Ullrich, P; O’Brien, T; Leung, LR; Ralph, FM; Wehner, M; Brands, S; Collow, A; Goldenson, N; Gorodetskaya, I; Griffith, H; Kashinath, K; Kawzenuk, B; Krishnan, H; Kurlin, V; Lavers, D; Magnusdottir, G; Mahoney, K; McClenny, E; Muszynski, G; Nguyen, PD; Prabhat, M; Qian, Y; Ramos, AM; Sarangi, C; Sellars, S; Shulgina, T; Tome, R; Waliser, D; Walton, D; Wick, G; Wilson, AM; Viale, M | Abstract: Atmospheric rivers (ARs) are now widely known for their association with high-impact weather events and long-term water supply in many regions. Researchers within the scientific community have developed numerous methods to identify and track of ARs—a necessary step for analyses on gridded data sets, and objective attribution of impacts to ARs. These different methods have been developed to answer specific research questions and hence use different criteria (e.g., geometry, threshold values of key variables, and time dependence). Furthermore, these methods are often employed using different reanalysis data sets, time periods, and regions of interest. The goal of the Atmospheric River Tracking Method Intercomparison Project (ARTMIP) is to understand and quantify uncertainties in AR science that arise due to differences in these methods. This paper presents results for key AR-related metrics based on 20+ different AR identification and tracking methods applied to Modern-Era Retrospective Analysis for Research and Applications Version 2 reanalysis data from January 1980 through June 2017. We show that AR frequency, duration, and seasonality exhibit a wide range of results, while the meridional distribution of these metrics along selected coastal (but not interior) transects are quite similar across methods. Furthermore, methods are grouped into criteria-based clusters, within which the range of results is reduced. AR case studies and an evaluation of individual method deviation from an all-method mean highlight advantages/disadvantages of certain approaches. For example, methods with less (more) restrictive criteria identify more (less) ARs and AR-related impacts. Finally, this paper concludes with a discussion and recommendations for those conducting AR-related research to consider.

In 1833 Gauss defined the linking number of two disjoint curves in 3-space. For open curves this double integral over the parameterised curves is real-valued and invariant modulo rigid motions or isometries that preserve distances between points, and has been recently used in the elucidation of molecular structures. In 1976 Banchoff geometrically interpreted the linking number between two line segments. An explicit analytic formula based on this interpretation was given in 2000 without proof in terms of 6 isometry invariants: the distance and angle between the segments and 4 coordinates specifying their relative positions. We give a detailed proof of this formula and describe its asymptotic behaviour that wasn’t previously studied.

We develop discrete geometry methods to resolve the data ambiguity challenge for periodic point sets to accelerate materials discovery. In any high-dimensional Euclidean space, a periodic point set is obtained from a finite set (motif) of points in a parallelepiped (unit cell) by periodic translations of the motif along basis vectors of the cell.

It is a core problem in any field to reliably tell how close two objects are to being the same, and once this relation has been established we can use this information to precisely quantify potential relationships, both analytically and with machine learning (ML). For inorganic solids, the chemical composition is a fundamental descriptor, which can be represented by assigning the ratio of each element in the material to a vector. These vectors are a convenient mathematical data structure for measuring similarity, but unfortunately, the standard metric (the Euclidean distance) gives little to no variance in the resultant distances between chemically dissimilar compositions. We present the Earth Mover’s Distance (EMD) for inorganic compositions, a well-defined metric which enables the measure of chemical similarity in an explainable fashion. We compute the EMD between two compositions from the ratio of each of the elements and the absolute distance between the elements on the modified Pettifor scale. This simple metric shows clear strength at distinguishing compounds and is efficient to compute in practice. The resultant distances have greater alignment with chemical understanding than the Euclidean distance, which is demonstrated on the binary compositions of the Inorganic Crystal Structure Database (ICSD). The EMD is a reliable numeric measure of chemical similarity that can be incorporated into automated workflows for a range of ML techniques. We have found that with no supervision the use of this metric gives a distinct partitioning of binary compounds into clear trends and families of chemical property, with future applications for nearest neighbor search queries in chemical database retrieval systems and supervised ML techniques.

It is a core problem in any field to reliably tell how close two objects are to being the same, and once this relation has been established we can use this information to precisely quantify potential relationships, both analytically and with machine learning (ML). For inorganic solids, the chemical composition is a fundamental descriptor, which can be represented by assigning the ratio of each element in the material to a vector. These vectors are a convenient mathematical data structure for measuring similarity, but unfortunately, the standard metric (the Euclidean distance) gives little to no variance in the resultant distances between chemically dissimilar compositions. We present the Earth Mover’s Distance (EMD) for inorganic compositions, a well-defined metric which enables the measure of chemical similarity in an explainable fashion. We compute the EMD between two compositions from the ratio of each of the elements and the absolute distance between the elements on the modified Pettifor scale. This simple metric shows clear strength at distinguishing compounds and is efficient to compute in practice. The resultant distances have greater alignment with chemical understanding than the Euclidean distance, which is demonstrated on the binary compositions of the Inorganic Crystal Structure Database (ICSD). The EMD is a reliable numeric measure of chemical similarity that can be incorporated into automated workflows for a range of ML techniques. We have found that with no supervision the use of this metric gives a distinct partitioning of binary compounds into clear trends and families of chemical property, with future applications for nearest neighbor search queries in chemical database retrieval systems and supervised ML techniques.

The tree reconstruction problem is to find an embedded straight-line tree that approximates a given cloud of unorganized points in $\mathbb{R}^m$ up to a certain error. A practical solution to this problem will accelerate a discovery of new colloidal products with desired physical properties such as viscosity. We define the Approximate Skeleton of any finite point cloud $C$ in a Euclidean space with theoretical guarantees. The Approximate Skeleton ASk$(C)$ always belongs to a given offset of $C$, i.e. the maximum distance from $C$ to ASk$(C)$ can be a given maximum error. The number of vertices in the Approximate Skeleton is close to the minimum number in an optimal tree by factor 2. The new Approximate Skeleton of any unorganized point cloud $C$ is computed in a near linear time in the number of points in $C$. Finally, the Approximate Skeleton outperforms past skeletonization algorithms on the size and accuracy of reconstruction for a large dataset of real micelles and random clouds., Accepted to topoInvis conference

This paper develops a new continuous approach to a similarity between periodic lattices of ideal crystals. Quantifying a similarity between crystal structures is needed to substantially speed up the Crystal Structure Prediction, because the prediction of many target properties of crystal structures is computationally slow and is essentially repeated for many nearly identical simulated structures. The proposed distances between arbitrary periodic lattices of crystal structures are invariant under all rigid motions, satisfy the metric axioms and continuity under atomic perturbations. The above properties make these distances ideal tools for clustering and visualizing large datasets of crystal structures. All the conclusions are rigorously proved and justified by experiments on real and simulated crystal structures reported in the Nature 2017 paper "Functional materials discovery using energy-structure-function maps".

Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data – for example, to approximate a point cloud by a low-dimensional non-linear subspace such as an embedded graph or a simplicial complex. Classical clustering methods and principal component analysis work well when data points split into good clusters or lie near linear subspaces of a Euclidean space. Methods from topological data analysis in general metric spaces detect more complicated patterns such as holes and voids that persist for a large interval in a 1-parameter family of shapes associated to a cloud. These features can be visualized in the form of a 1-dimensional homologically persistent skeleton, which optimally extends a minimum spanning tree of a point cloud to a graph with cycles. We generalize this skeleton to higher dimensions and prove its optimality among all complexes that preserve topological features of data at any scale.

The over-segmentation problem is to split a pixel-based image into a smaller number of superpixels that can be treated as indecompasable regions to speed up higher level image processing such as segmentation or object detection. A traditional superpixel is a potentially disconnected union of square pixels, which can have complicated topology (with holes) and geometry (highly zigzag boundaries). This paper contributes to new resolution-independent superpixels modeled as convex polygons with straight-line edges and vertices with real coordinates not restricted to a fixed pixel grid. Any such convex polygon can be rendered at any resolution higher than in original images, hence superpixels are resolution-independent. The key difficulty in obtaining resolution-independent superpixels is to find continuous straight-line edges, while classical edge detection focuses on extracting only discrete edge pixels. The recent Persistent Line Segment Detector (PLSD) avoids intersections and small angles between line segments, which are hard to fix before a proper polygonal mesh can be constructed. The key novelty is an automatic selection of strongest straight-line segments by using the concept of persistence from Topological Data Analysis, which allows to rank segments by their strength. The PLSD performed well in comparison with the only past Line Segment Detector Algorithm (LSDA) on the Berkeley Segmentation Database of 500 real-life images. The PLSD is now extended to the Persistent Resolution-Independent Mesh (PRIM).

Data Science aims to extract meaningful knowledge from unorganised data. Real datasets usually come in the form of a cloud of points with only pairwise distances. Numerous applications require to visualise an overall shape of a noisy cloud of points sampled from a non-linear object that is more complicated than a union of disjoint clusters. The skeletonisation problem in its hardest form is to find a 1-dimensional skeleton that correctly represents a shape of the cloud. This paper compares several algorithms that solve the above skeletonisation problem for any point cloud and guarantee a successful reconstruction. For example, given a highly noisy point sample of an unknown underlying graph, a reconstructed skeleton should be geometrically close and homotopy equivalent to (has the same number of independent cycles as) the underlying graph. One of these algorithm produces a Homologically Persistent Skeleton (HoPeS) for any cloud without extra parameters. This universal skeleton contains sub-graphs that provably represent the 1-dimensional shape of the cloud at any scale. Other subgraphs of HoPeS reconstruct an unknown graph from its noisy point sample with a correct homotopy type and within a small offset of the sample. The extensive experiments on synthetic and real data reveal for the first time the maximum level of noise that allows successful graph reconstructions., Comment: This paper has been published in the journal Pattern Recognition

The over-segmentation into superpixels is an important pre-processing step to smartly compress the input size and speed up higher level tasks. A superpixel was traditionally considered as a small cluster of square-based pixels that have similar color intensities and are closely located to each other. In this discrete model the boundaries of superpixels often have irregular zigzags consisting of horizontal or vertical edges from a given pixel grid. However digital images represent a continuous world, hence the following continuous model in the resolution-independent formulation can be more suitable for the reconstruction problem.

Edge detection algorithms usually produce a discrete set of edgels (edge pixels) in a given image on a fixed pixel grid. We consider the harder problem of detecting continuous straight line segments at subpixel resolution. The state-of-the art Line Segment Detection Algorithm (LSDA) outputs unordered line segments whose total number cannot be easily controlled. Another motivation to improve the LSDA is to avoid intersections and small angles between line segments, hence difficulties in higher level tasks such as segmentation or contour extraction.

Author(s): Shields, Christine A; Rutz, Jonathan J; Leung, Lai-Yung; Ralph, F Martin; Wehner, Michael; Kawzenuk, Brian; Lora, Juan M; McClenny, Elizabeth; Osborne, Tashiana; Payne, Ashley E; Ullrich, Paul; Gershunov, Alexander; Goldenson, Naomi; Guan, Bin; Qian, Yun; Ramos, Alexandre M; Sarangi, Chandan; Sellars, Scott; Gorodetskaya, Irina; Kashinath, Karthik; Kurlin, Vitaliy; Mahoney, Kelly; Muszynski, Grzegorz; Pierce, Roger; Subramanian, Aneesh C; Tome, Ricardo; Waliser, Duane; Walton, Daniel; Wick, Gary; Wilson, Anna; Lavers, David; Collow, Allison; Krishnan, Harinarayan; Magnusdottir, Gudrun; Nguyen, Phu

Image over-segmentation is formalized as the approximation problem when a large image is segmented into a small number of connected superpixels with best fitting colors. The approximation quality is measured by the energy whose main term is the sum of squared color deviations over all pixels and a regularizer encourages round shapes. The first novelty is the coarse initialization of a non-uniform superpixel mesh based on selecting most persistent edge segments. The second novelty is the scale-invariant regularizer based on the isoperimetric quotient. The third novelty is the improved coarse-to-fine optimization where local moves are organized according to their energy improvements. The algorithm beats the state-of-the-art on the objective reconstruction error and performs similarly to other superpixels on the benchmarks of BSD500. The only parameters are the number of superpixels and the shape coefficient for a trade-off between accuracy and shape of superpixels

We design a fast algorithm for computing the fundamental group of the complement to any knotted polygonal graph in 3-space. A polygonal graph consists of straight segments and is given by sequences of vertices along edge-paths. This polygonal model is motivated by protein backbones described in the Protein Data Bank by 3D positions of atoms. Our KGG algorithm simplifies a knotted graph and computes a short presentation of the Knotted Graph Group containing powerful invariants for classifying graphs up to isotopy. We use only a reduced plane diagram without building a large complex representing the complement of a graph in 3-space.

We introduce simple codes and fast visualization tools for knotted structures in complicated molecules and brain networks. Knots, links and more general knotted graphs are studied up to an ambient isotopy in Euclidean 3-space. A knotted graph can be represented by a plane diagram or a Gauss code. First we recognize in linear time if an abstract Gauss code represents a graph embedded in 3-space. Second we design a fast algorithm for drawing any knotted graph in the 3-page book, which is a union of 3 half-planes along their common boundary. The complexity of the algorithm is linear in the length of a Gauss code. Three-page embeddings are encoded in such a way that the isotopy classification for graphs in 3-space reduces to a word problem in finitely presented semigroups.

Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic images of link groups., Comment: 22 pages, 4 figures, examples added

Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations., 8 pages => 5 pages of main text plus 3 pages in appendix. Submitted to CCCG 2015

Construction of a semigroup with 15 generators and 84 relations is given. The center of this semigroup is in one-to-one correspondence with the set of all isotopy classes of non-oriented singular knots (links with finitely many double intersections in general position) in three-dimensional space., Comment: 14 pages, 5 figures

A connected metric graph G with n vertices and without loops and multiple edges is given as an n × n-matrix whose entry aij is the length of a single edge between vertices i ≠ j. A robot in the metric graph G is the metric ball with a center x ϵ G and a radius r > 0. The configuration space OC(G, r) of 2 ordered robots in G is the set of all centers (x, y)ϵ G×G such that x, y are at least 2r away from each other. We introduce the configuration skeleton CS(G, r) ⊂ OC(G, r) that captures all connectivity information of the larger space OC(G, r). We design an algorithm of time complexity O(n2) to find all connected components of OC(G, r) that are maximal subsets of all safe positions (x, y) connectable by collision-free motions of the two round robots.

Recently, Dynnikov suggested a new way of encoding nonoriented links by three-page diagrams [1], and this permitted him to obtain an algebraic classification of the isotopy classes of nonoriented links (see [2]). We generalize this approach to 3-valent graphs. A finite graph is said to be 3-valent if exactly three edges meet at any vertex. We consider only 3-valent nonoriented graphs, which can be disconnected and have loops and multi-edges. By a spatial graph we mean an embedding of a graph in R3 under which the edges become finite polygonal lines. We study the spatial graphs up to an ambient isotopy, where an ambient isotopy between two graphs is a continuous family of homeomorphisms φt : R3 → R3 , t ∈ [0, 1], such that φ0 = id and φ1 sends one of these graphs to the other. Isotopy invariants of spatial graphs were studied in [3, 4]. For other equivalence relations (including concordance, homotopy, and homology), see [5]. The notion of spatial graph is motivated both theoretically and practically. First, the problem to classify the spatial graphs up to isotopy is a special case of the general topological problem of classifying the embeddings in Euclidean space. We simultaneously obtain a natural extension of the classical knot theory to more complex one-dimensional objects. Many invariants of ordinary links, including the Alexander polynomials and Vassiliev invariants, can be generalized to graphs [6, 7]. As well as the ordinary links, the spatial graphs can readily be represented by plane diagrams determined up to the following Reidemeister moves

Let a point cloud be a noisy dotted image of a graph on the plane. We present a new fast algorithm for reconstructing the original graph from the given point cloud. Degrees of vertices in the graph are found by methods of persistent topology. Necessary parameters are automatically optimized by machine learning tools.