arXiv subjects

Vitaliy Kurlin

Publications and source records attributed to Vitaliy Kurlin.

At least 19 recordsLinked to original sources

arXiv2026

Complete invariants of atomic clouds under rigid motion with Lipschitz continuous metrics in a polynomial time

A basic representation of any real molecule is a finite cloud of unordered atoms, many of which are chemically indistinguishable. A natural equivalence on point clouds in any metric space is defined by isometries that are distance-preserving transformations. In a Euclidean space, any isometry is a composition of translations, rotations, and reflections. If points are ordered, the isometry class of this cloud is uniquely determined by the matrix of all pairwise distances. If m points are unordered, a naive metric based on distance matrices needs exponentially many m! permutations. We define a complete invariant for n-dimensional clouds of m unordered points under rigid motion, which distinguishes all mirror images in R^n. The key challenge was to design a distance on invariant values that is Lipschitz continuous under noise and computable in a polynomial time of cloud sizes, for a fixed dimension n.

math.MG↗

arXiv2026

The strength of a geometric simplex

The basic input for many real objects is a finite cloud of unordered points. The strongest equivalence between objects in practice is rigid motion in a Euclidean space. A recent polynomial-time classification of point clouds required a Lipschitz continuous function that vanishes on degenerate simplices, while the usual volume is not Lipschitz. We define the strength of any geometric simplex and prove its continuity under perturbations with explicit bounds for Lipschitz constants.

math.MG↗

arXiv2026

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

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

math.MG↗

arXiv2025

Computing the bridge length: the key ingredient in a continuous isometry classification of periodic point sets

The fundamental model of any periodic crystal is a periodic set of points at all atomic centres. Since crystal structures are determined in a rigid form, their strongest equivalence is rigid motion (composition of translations and rotations) or isometry (also including reflections). The recent classification of periodic point sets under rigid motion used a complete invariant isoset whose size essentially depends on the bridge length, defined as the minimum `jump' that suffices to connect any points in the given set. We propose a practical algorithm to compute the bridge length of any periodic point set given by a motif of points in a periodically translated unit cell. The algorithm has been tested on a large crystal dataset and is required for an efficient continuous classification of all periodic crystals. The exact computation of the bridge length is a key step to realising the inverse design of materials from new invariant values.

cs.CG↗

arXiv2025

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

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

cs.CG↗

arXiv2025

Complete and bi-continuous invariant of protein backbones under rigid motion

Proteins are large biomolecules that regulate all living organisms and consist of one or several chains. The primary structure of a protein chain is a sequence of amino acid residues whose three main atoms (alpha-carbon, nitrogen, and carbonyl carbon) form a protein backbone. The tertiary structure is the rigid shape of a protein chain represented by atomic positions in 3-dimensional space. Because different geometric structures often have distinct functional properties, it is important to continuously quantify differences in rigid shapes of protein backbones. Unfortunately, many widely used similarities of proteins fail axioms of a distance metric and discontinuously change under tiny perturbations of atoms. This paper develops a complete invariant that identifies any protein backbone in 3-dimensional space, uniquely under rigid motion. This invariant is Lipschitz bi-continuous in the sense that it changes up to a constant multiple of a maximum perturbation of atoms, and vice versa. The new invariant has been used to detect thousands of (near-)duplicates in the Protein Data Bank, whose presence inevitably skews machine learning predictions. The resulting invariant space allows low-dimensional maps with analytically defined coordinates that reveal substantial variability in the protein universe.

cs.CG↗

arXiv2025

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

This paper rigorously solves the challenging problem of recognizing periodic patterns under rigid motion in Euclidean geometry. The 3-dimensional case is practically important for justifying the novelty of solid crystalline materials (periodic crystals) and for patenting medical drugs in a solid tablet form. Past descriptors based on finite subsets fail when a unit cell of a periodic pattern discontinuously changes under almost any perturbation of atoms, which is inevitable due to noise and atomic vibrations. The major problem is not only to find complete invariants (descriptors with no false negatives and no false positives for all periodic patterns) but to design efficient algorithms for distance metrics on these invariants that should continuously behave under noise. The proposed continuous metrics solve this problem in any Euclidean dimension and are algorithmically approximated with small error factors in times that are explicitly bounded in the size and complexity of a given pattern. The proved Lipschitz continuity allows us to confirm all near-duplicates filtered by simpler invariants in major databases of experimental and simulated crystals. This practical detection of noisy duplicates will stop the artificial generation of `new' materials from slight perturbations of known crystals. Several such duplicates are under investigation by five journals for data integrity.

math.MG↗

arXiv2024

The importance of definitions in crystallography

This paper was motivated by the articles "Same or different - that is the question" in CrystEngComm (July 2020) and "Change to the definition of a crystal" in the IUCr newsletter (June 2021). Experimental approaches to crystal comparisons require rigorously defined classifications in crystallography and beyond. Since crystal structures are determined in a rigid form, their strongest equivalence in practice is rigid motion, which is a composition of translations and rotations in 3-dimensional space. Conventional representations based on reduced cells and standardizations theoretically distinguish all periodic crystals. However, all cell-based representations are inherently discontinuous under almost any atomic displacement that can arbitrarily scale up a reduced cell. Hence comparing millions of known structures in materials databases needs continuous distance metrics.

cond-mat.mtrl-sci↗

arXiv2024

Generic families of finite metric spaces with identical or trivial 1-dimensional persistence

Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of persistent homology provides an upper bound for the change of persistence in the bottleneck distance under perturbations of points, but without giving a lower bound. This paper clarifies the possible limitations persistent homology may have in distinguishing finite metric spaces, which is evident for non-isometric point sets with identical persistence. We describe generic families of point sets in metric spaces that have identical or even trivial one-dimensional persistence. The results motivate stronger invariants to distinguish finite point sets up to isometry.

cs.CG↗

arXiv2024

Material Property Prediction using Graphs based on Generically Complete Isometry Invariants

The structure-property hypothesis says that the properties of all materials are determined by an underlying crystal structure. The main obstacle was the ambiguity of conventional crystal representations based on incomplete or discontinuous descriptors that allow false negatives or false positives. This ambiguity was resolved by the ultra-fast Pointwise Distance Distribution (PDD), which distinguished all periodic structures in the world's largest collection of real materials (Cambridge Structural Database). The state-of-the-art results in property predictions were previously achieved by graph neural networks based on various graph representations of periodic crystals, including the Crystal Graph with vertices at all atoms in a crystal unit cell. This work adapts the Pointwise Distance Distribution for a simpler graph whose vertex set is not larger than the asymmetric unit of a crystal structure. The new Distribution Graph reduces mean-absolute-error by 0.6\%-12\% while having 44\%-88\% of the number of vertices when compared to the crystal graph when applied on the Materials Project and Jarvis-DFT datasets using CGCNN and ALIGNN. Methods for hyper-parameters selection for the graph are backed by the theoretical results of the Pointwise Distance Distribution and are then experimentally justified.

physics.comp-ph↗

arXiv2024

Accelerating Material Property Prediction using Generically Complete Isometry Invariants

Periodic material or crystal property prediction using machine learning has grown popular in recent years as it provides a computationally efficient replacement for classical simulation methods. A crucial first step for any of these algorithms is the representation used for a periodic crystal. While similar objects like molecules and proteins have a finite number of atoms and their representation can be built based upon a finite point cloud interpretation, periodic crystals are unbounded in size, making their representation more challenging. In the present work, we adapt the Pointwise Distance Distribution (PDD), a continuous and generically complete isometry invariant for periodic point sets, as a representation for our learning algorithm. The PDD distinguished all (more than 660 thousand) periodic crystals in the Cambridge Structural Database as purely periodic sets of points without atomic types. We develop a transformer model with a modified self-attention mechanism that combines PDD with compositional information via a spatial encoding method. This model is tested on the crystals of the Materials Project and Jarvis-DFT databases and shown to produce accuracy on par with state-of-the-art methods while being several times faster in both training and prediction time.

cs.LG↗

arXiv2024

A new near-linear time algorithm for k-nearest neighbor search using a compressed cover tree

Given a reference set $R$ of $n$ points and a query set $Q$ of $m$ points in a metric space, this paper studies an important problem of finding $k$-nearest neighbors of every point $q \in Q$ in the set $R$ in a near-linear time. In the paper at ICML 2006, Beygelzimer, Kakade, and Langford introduced a cover tree on $R$ and attempted to prove that this tree can be built in $O(n\log n)$ time while the nearest neighbor search can be done in $O(n\log m)$ time with a hidden dimensionality factor. This paper fills a substantial gap in the past proofs of time complexity by defining a simpler compressed cover tree on the reference set $R$. The first new algorithm constructs a compressed cover tree in $O(n \log n)$ time. The second new algorithm finds all $k$-nearest neighbors of all points from $Q$ using a compressed cover tree in time $O(m(k+\log n)\log k)$ with a hidden dimensionality factor depending on point distributions of the given sets $R,Q$ but not on their sizes.

cs.CG↗

arXiv2023

Polynomial-time algorithms for continuous metrics on atomic clouds of unordered points

The most fundamental model of a molecule is a cloud of unordered atoms, even without chemical bonds that can depend on thresholds for distances and angles. The strongest equivalence between clouds of atoms is rigid motion, which is a composition of translations and rotations. The existing datasets of experimental and simulated molecules require a continuous quantification of similarity in terms of a distance metric. While clouds of m ordered points were continuously classified by Lagrange's quadratic forms (distance matrices or Gram matrices), their extensions to m unordered points are impractical due to the exponential number of m! permutations. We propose new metrics that are continuous in general position and are computable in a polynomial time in the number m of unordered points in any Euclidean space of a fixed dimension n.

cs.CG↗

arXiv2023

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

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

cs.CV↗

arXiv2023

Simplexwise Distance Distributions for finite spaces with metrics and measures

A finite set of unlabelled points in Euclidean space is the simplest representation of many real objects from mineral rocks to sculptures. Since most solid objects are rigid, their natural equivalence is rigid motion or isometry maintaining all inter-point distances. More generally, any finite metric space is an example of a metric-measure space that has a probability measure and a metric satisfying all axioms. This paper develops Simplexwise Distance Distributions (SDDs) for any finite metric spaces and metric-measures spaces. These SDDs classify all known non-equivalent spaces that were impossible to distinguish by simpler invariants. We define metrics on SDDs that are Lipschitz continuous and allow exact computations whose parametrised complexities are polynomial in the number of given points.

math.MG↗

arXiv2023

Entropic trust region for densest crystallographic symmetry group packings

Molecular crystal structure prediction (CSP) seeks the most stable periodic structure given the chemical composition of a molecule and pressure-temperature conditions. Modern CSP solvers use global optimization methods to search for structures with minimal free energy within a complex energy landscape induced by intermolecular potentials. A major caveat of these methods is that initial configurations are random, making thus the search susceptible to convergence at local minima. Providing initial configurations that are densely packed with respect to the geometric representation of a molecule can significantly accelerate CSP. Motivated by these observations, we define a class of periodic packings restricted to crystallographic symmetry groups (CSG) and design a search method for the densest CSG packings in an information-geometric framework. Since the CSG induce a toroidal topology on the configuration space, a non-Euclidean trust region method is performed on a statistical manifold consisting of probability distributions defined on an $n$-dimensional flat unit torus by extending the multivariate von Mises distribution. Introducing an adaptive quantile reformulation of the fitness function into the optimization schedule provides the algorithm with a geometric characterization through local dual geodesic flows. Moreover, we examine the geometry of the adaptive selection-quantile defined trust region and show that the algorithm performs a maximization of stochastic dependence among elements of the extended multivariate von Mises distributed random vector. We experimentally evaluate the behavior and performance of the method on various densest packings of convex polygons in $2$-dimensional CSGs for which optimal solutions are known, and demonstrate its application in the pentacene thin-film CSP.

math.OC↗

arXiv2023

Density functions of periodic sequences of continuous events

Periodic Geometry studies isometry invariants of periodic point sets that are also continuous under perturbations. The motivations come from periodic crystals whose structures are determined in a rigid form but any minimal cells can discontinuously change due to small noise in measurements. For any integer k>=0, the density function of a periodic set S was previously defined as the fractional volume of all k-fold intersections (within a minimal cell) of balls that have a variable radius t and centers at all points of S. This paper introduces the density functions for periodic sets of points with different initial radii motivated by atomic radii of chemical elements and by continuous events occupying disjoint intervals in time series. The contributions are explicit descriptions of the densities for periodic sequences of intervals. The new densities are strictly stronger and distinguish periodic sequences that have identical densities in the case of zero radii.

cs.CG↗

arXiv2022

Densest plane group packings of regular polygons

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. Former results were mainly regarding densest lattice or double-lattice configurations. Here we consider all two-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 two-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 means of the Entropic Trust Region Packing Algorithm that approximately solves this problem, we examine some known and unknown densest packings of various $n$-gons in all $17$ plane groups and state conjectures about common symmetries of the densest plane group packings for every $n$-gon.

math.MG↗