arXiv subjects

Herbert Edelsbrunner

Publications and source records attributed to Herbert Edelsbrunner.

At least 37 records · Page 2Linked to original sources

arXiv2024

Brillouin Zones of Integer Lattices and Their Perturbations

For a locally finite set, $A \subseteq \mathbb{R}^d$, the $k$-th Brillouin zone of $a \in A$ is the region of points $x \in \mathbb{R}^d$ for which $\|x-a\|$ is the $k$-th smallest among the Euclidean distances between $x$ and the points in $A$. If $A$ is a lattice, the $k$-th Brillouin zones of the points in $A$ are translates of each other, which tile space. Depending on the value of $k$, they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in $\mathbb{R}^2$, and the convergence of the maximum volume of a chamber to zero for the integer lattice.

math.CO↗

arXiv2023

Maximum Betti numbers of Čech complexes

The Upper Bound Theorem for convex polytopes implies that the $p$-th Betti number of the Čech complex of any set of $N$ points in $\mathbb R^d$ and any radius satisfies $β_{p} = O(N^{m})$, with $m = \min \{ p+1, \lceil d/2 \rceil \}$. We construct sets in even and odd dimensions that prove this upper bound is asymptotically tight. For example, we describe a set of $N = 2(n+1)$ points in $\mathbb R^3$ and two radii such that the first Betti number of the Čech complex at one radius is $(n+1)^2 - 1$, and the second Betti number of the Čech complex at the other radius is $n^2$.

math.CO↗

arXiv2022

On the Size of Chromatic Delaunay Mosaics

Given a locally finite set $A \subseteq \mathbb{R}^d$ and a coloring $χ\colon A \to \{0,1,\ldots,s\}$, we introduce the chromatic Delaunay mosaic of $χ$, which is a Delaunay mosaic in $\mathbb{R}^{s+d}$ that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that $d$ and $s$ are constants. For example, if $A$ is finite with $n = \#{A}$, and the coloring is random, then the chromatic Delaunay mosaic has $O(n^{\lceil{d/2}\rceil})$ cells in expectation. In contrast, for Delone sets and Poisson point processes in $\mathbb{R}^d$, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in $\mathbb{R}^2$ all colorings of a dense set of $n$ points have chromatic Delaunay mosaics of size $O(n)$. This encourages the use of chromatic Delaunay mosaics in applications.

math.CO↗

arXiv2022

On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane

For a locally finite set in $\mathbb{R}^2$, the order-$k$ Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in $k$. As an example, a stationary Poisson point process in $\mathbb{R}^2$ is locally finite, coarsely dense, and generic with probability one. For such a set, the distribution of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-$1$ Delaunay mosaics given by Miles in 1970.

math.CO↗

arXiv2021

The Density Fingerprint of a Periodic Point Set

Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction.

cs.CG↗

arXiv2021

Tri-Partitions and Bases of an Ordered Complex

Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholz-Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, $K$, and every dimension, $p$, there is a partition of the set of $p$-cells into a maximal $p$-tree, a maximal $p$-cotree, and a collection of $p$-cells whose cardinality is the $p$-th Betti number of $K$. Given an ordering of the $p$-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.

math.CO↗

arXiv2020

A Simple Algorithm for Higher-order Delaunay Mosaics and Alpha Shapes

We present a simple algorithm for computing higher-order Delaunay mosaics that works in Euclidean spaces of any finite dimensions. The algorithm selects the vertices of the order-$k$ mosaic from incrementally constructed lower-order mosaics and uses an algorithm for weighted first-order Delaunay mosaics as a black-box to construct the order-$k$ mosaic from its vertices. Beyond this black-box, the algorithm uses only combinatorial operations, thus facilitating easy implementation. We extend this algorithm to compute higher-order $α$-shapes and provide open-source implementations. We present experimental results for properties of higher-order Delaunay mosaics of random point sets.

cs.CG↗

arXiv2020

The Weighted Gaussian Curvature Derivative of a Space-Filling Diagram

The morphometric approach [HRC13,RHK06] writes the solvation free energy as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted Gaussian curvature. Together with the derivatives of the weighted volume in [EdKo03], the weighted area in [BEKL04], and the weighted mean curvature in [AkEd19], this yields the derivative of the morphometric expression of solvation free energy.

cs.CG↗

arXiv2020

The Beauty of Random Polytopes Inscribed in the 2-sphere

Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case $d=3$, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.

math.MG↗

arXiv2020

The Weighted Mean Curvature Derivative of a Space-Filling Diagram

Representing an atom by a solid sphere in $3$-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free energy. The morphometric approach [HRC13,RHK06] writes the latter as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted mean curvature. Together with the derivatives of the weighted volume in [EdKo03], the weighted area in [BEKL04], and the weighted Gaussian curvature [AkEd19], this yields the derivative of the morphometric expression of the solvation free energy.

cs.CG↗

arXiv2020

Čech-Delaunay gradient flow and homology inference for self-maps

We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspace of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for Čech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive Čech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems.

math.AT↗

arXiv2019

Weighted Poisson-Delaunay Mosaics

Slicing a Voronoi tessellation in $\mathbb{R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in $\mathbb{R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a byproduct, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in $\mathbb{R}^n$

math.PR↗

arXiv2019

On the Configuration Space of Steiner Minimal Trees

Among other results, we prove the following theorem about Steiner minimal trees in $d$-dimensional Euclidean space: if two finite sets in $\mathbb{R}^d$ have unique and combinatorially equivalent Steiner minimal trees, then there is a homotopy between the two sets that maintains the uniqueness and the combinatorial structure of the Steiner minimal tree throughout the homotopy.

math.MG↗

arXiv2019

Topological Data Analysis in Information Space

Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable usage of existing computational topology software in this context.

cs.CG↗

arXiv2018

Unexpected Topology of the Temperature Fluctuations in the Cosmic Microwave Background

We study the topology generated by the temperature fluctuations of the Cosmic Microwave Background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the Planck satellite with a thousand simulated maps generated according to the LCDM paradigm with Gaussian distributed fluctuations. The survey of the CMB over $\mathbb{S}^2$ is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of "masks" is of importance, we introduce the concept of relative homology. The parametric $χ^2$-test shows differences between observations and simulations, yielding $p$-values at per-cent to less than per-mil levels roughly between 2 to 7 degrees. The highest observed deviation for $b_0$ and $b_1$ is approximately between $3σ$-4$σ$ at scales of 3 to 7 degrees. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66 degrees in the literature, computed from independent measurements of the CMB temperature fluctuations by Planck's predecessor WMAP satellite. The mildly anomalous behaviour of Euler characteristic is related to the strongly anomalous behaviour of components and holes. These are also the scales at which the observed maps exhibit low variance compared to the simulations. Non-parametric tests show even stronger differences at almost all scales. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the super-horizon scales involved, may motivate to look at primordial non-Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology.

astro-ph.CO↗

arXiv2017

Poisson-Delaunay Mosaics of Order $k$

The order-$k$ Voronoi tessellation of a locally finite set $X \subseteq \mathbb{R}^n$ decomposes $\mathbb{R}^n$ into convex domains whose points have the same $k$ nearest neighbors in $X$. Assuming $X$ is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the $k$ nearest points in $X$ are within a given distance threshold.

math.PR↗

arXiv2017

Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics

Using the geodesic distance on the $n$-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. Assuming the points are not contained in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of the convex hull in $\mathbb{R}^{n+1}$, so we also get the expected number of faces of a random inscribed polytope. We find that the expectations are essentially the same as for the Poisson-Delaunay mosaic in $n$-dimensional Euclidean space. As proved by Antonelli and collaborators, an orthant section of the $n$-sphere is isometric to the standard $n$-simplex equipped with the Fisher information metric. It follows that the latter space has similar stochastic properties as the $n$-dimensional Euclidean space. Our results are therefore relevant in information geometry and in population genetics.

math.PR↗

arXiv2017

The Topology of the Cosmic Web in Terms of Persistent Betti Numbers

We introduce a multiscale topological description of the Megaparsec weblike cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multiscale arrangement of matter and galaxies. Emanating from algebraic topology and Morse theory, Betti numbers and persistence diagrams represent an extension and deepening of the cosmologically familiar topological genus measure, and the related geometric Minkowski functionals. In addition to a description of the mathematical background, this study presents the computational procedure for computing Betti numbers and persistence diagrams for density field filtrations. The field may be computed starting from a discrete spatial distribution of galaxies or simulation particles. The main emphasis of this study concerns an extensive and systematic exploration of the imprint of different weblike morphologies and different levels of multiscale clustering in the corresponding computed Betti numbers and persistence diagrams. To this end, we use Voronoi clustering models as templates for a rich variety of weblike configurations, and the fractal-like Soneira-Peebles models exemplify a range of multiscale configurations. We have identified the clear imprint of cluster nodes, filaments, walls, and voids in persistence diagrams, along with that of the nested hierarchy of structures in multiscale point distributions. We conclude by outlining the potential of persistent topology for understanding the connectivity structure of the cosmic web, in large simulations of cosmic structure formation and in the challenging context of the observed galaxy distribution in large galaxy surveys.

astro-ph.CO↗