arXiv subjects

Michael Kerber

Publications and source records attributed to Michael Kerber.

At least 55 records · Page 3Linked to original sources

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↗

arXiv2016

Geometry Helps to Compare Persistence Diagrams

Exploiting geometric structure to improve the asymptotic complexity of discrete assignment problems is a well-studied subject. In contrast, the practical advantages of using geometry for such problems have not been explored. We implement geometric variants of the Hopcroft--Karp algorithm for bottleneck matching (based on previous work by Efrat el al.) and of the auction algorithm by Bertsekas for Wasserstein distance computation. Both implementations use k-d trees to replace a linear scan with a geometric proximity query. Our interest in this problem stems from the desire to compute distances between persistence diagrams, a problem that comes up frequently in topological data analysis. We show that our geometric matching algorithms lead to a substantial performance gain, both in running time and in memory consumption, over their purely combinatorial counterparts. Moreover, our implementation significantly outperforms the only other implementation available for comparing persistence diagrams.

cs.CG↗

arXiv2016

Homotopy equivalence between Voronoi medusa and Delaunay medusa

We trace movements of certain points in space-time along their corresponding continuous path. We partition the space at every moment of time using alpha-Complexes, Voronoi medusa is then the collection or union of restricted Voronoi cells at every moment in time. We can imagine them as a four dimensional structure formed when three dimensional restricted Voronoi cells sweeps continuously through the extra dimension of time. Similarly Delaunay medusa is the collection of the corresponding Delaunay triangulations at each moment in time. In this article we prove that these two structures are homotopic.

cs.CG↗

arXiv2016

Polynomial-Sized Topological Approximations Using The Permutahedron

Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for $n$ points in $\mathbb{R}^d$, we obtain a $O(d)$-approximation with at most $n2^{O(d \log k)}$ simplices of dimension $k$ or lower. In conjunction with dimension reduction techniques, our approach yields a $O(\mathrm{polylog} (n))$-approximation of size $n^{O(1)}$ for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry. Building on the same geometric concept, we also present a lower bound result on the size of an approximate filtration: we construct a point set for which every $(1+ε)$-approximation of the Čech filtration has to contain $n^{Ω(\log\log n)}$ features, provided that $ε<\frac{1}{\log^{1+c} n}$ for $c\in(0,1)$.

cs.CG↗

arXiv2015

Approximation and Streaming Algorithms for Projective Clustering via Random Projections

Let $P$ be a set of $n$ points in $\mathbb{R}^d$. In the projective clustering problem, given $k, q$ and norm $ρ\in [1,\infty]$, we have to compute a set $\mathcal{F}$ of $k$ $q$-dimensional flats such that $(\sum_{p\in P}d(p, \mathcal{F})^ρ)^{1/ρ}$ is minimized; here $d(p, \mathcal{F})$ represents the (Euclidean) distance of $p$ to the closest flat in $\mathcal{F}$. We let $f_k^q(P,ρ)$ denote the minimal value and interpret $f_k^q(P,\infty)$ to be $\max_{r\in P}d(r, \mathcal{F})$. When $ρ=1,2$ and $\infty$ and $q=0$, the problem corresponds to the $k$-median, $k$-mean and the $k$-center clustering problems respectively. For every $0 < ε< 1$, $S\subset P$ and $ρ\ge 1$, we show that the orthogonal projection of $P$ onto a randomly chosen flat of dimension $O(((q+1)^2\log(1/ε)/ε^3) \log n)$ will $ε$-approximate $f_1^q(S,ρ)$. This result combines the concepts of geometric coresets and subspace embeddings based on the Johnson-Lindenstrauss Lemma. As a consequence, an orthogonal projection of $P$ to an $O(((q+1)^2 \log ((q+1)/ε)/ε^3) \log n)$ dimensional randomly chosen subspace $ε$-approximates projective clusterings for every $k$ and $ρ$ simultaneously. Note that the dimension of this subspace is independent of the number of clusters~$k$. Using this dimension reduction result, we obtain new approximation and streaming algorithms for projective clustering problems. For example, given a stream of $n$ points, we show how to compute an $ε$-approximate projective clustering for every $k$ and $ρ$ simultaneously using only $O((n+d)((q+1)^2\log ((q+1)/ε))/ε^3 \log n)$ space. Compared to standard streaming algorithms with $Ω(kd)$ space requirement, our approach is a significant improvement when the number of input points and their dimensions are of the same order of magnitude.

cs.CG↗

arXiv2015

Semi-dynamic connectivity in the plane

Motivated by a path planning problem we consider the following procedure. Assume that we have two points $s$ and $t$ in the plane and take $\mathcal{K}=\emptyset$. At each step we add to $\mathcal{K}$ a compact convex set that does not contain $s$ nor $t$. The procedure terminates when the sets in $\mathcal{K}$ separate $s$ and $t$. We show how to add one set to $\mathcal{K}$ in $O(1+kα(n))$ amortized time plus the time needed to find all sets of $\mathcal{K}$ intersecting the newly added set, where $n$ is the cardinality of $\mathcal{K}$, $k$ is the number of sets in $\mathcal{K}$ intersecting the newly added set, and $α(\cdot)$ is the inverse of the Ackermann function.

cs.CG↗

arXiv2015

The Offset Filtration of Convex Objects

We consider offsets of a union of convex objects. We aim for a filtration, a sequence of nested cell complexes, that captures the topological evolution of the offsets for increasing radii. We describe methods to compute a filtration based on the Voronoi partition with respect to the given convex objects. We prove that, in two and three dimensions, the size of the filtration is proportional to the size of the Voronoi diagram. Our algorithm runs in $Θ(n \log{n})$ in the $2$-dimensional case and in expected time $O(n^{3 + ε})$, for any $ε> 0$, in the $3$-dimensional case. Our approach is inspired by alpha-complexes for point sets, but requires more involved machinery and analysis primarily since Voronoi regions of general convex objects do not form a good cover. We show by experiments that our approach results in a similarly fast and topologically more stable method for computing a filtration compared to approximating the input by point samples.

cs.CG↗

arXiv2014

Local Doubling Dimension of Point Sets

We introduce the notion of t-restricted doubling dimension of a point set in Euclidean space as the local intrinsic dimension up to scale t. In many applications information is only relevant for a fixed range of scales. We present an algorithm to construct a hierarchical net-tree up to scale t which we denote as the net-forest. We present a method based on Locality Sensitive Hashing to compute all near neighbours of points within a certain distance. Our construction of the net-forest is probabilistic, and we guarantee that with high probability, the net-forest is supplemented with the correct neighbouring information. We apply our net-forest construction scheme to create an approximate Cech complex up to a fixed scale; and its complexity depends on the local intrinsic dimension up to that scale.

cs.CG↗

arXiv2014

Sphere Packing with Limited Overlap

The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as possible. We define a generalized version of the problem, where we allow each ball a limited amount of overlap with other balls. We study two natural choices of overlap measures and obtain the optimal lattice packings in a parameterized family of lattices which contains the FCC, BCC, and integer lattice.

cs.CG↗

arXiv2013

Distributed computation of persistent homology

Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not exhaust the available memory. Following up on a recently presented parallel method for persistence computation on shared memory systems, we demonstrate that a simple adaption of the standard reduction algorithm leads to a variant for distributed systems. Our algorithmic design ensures that the data is distributed over the nodes without redundancy; this permits the computation of much larger instances than on a single machine. Moreover, we observe that the parallelism at least compensates for the overhead caused by communication between nodes, and often even speeds up the computation compared to sequential and even parallel shared memory algorithms. In our experiments, we were able to compute the persistent homology of filtrations with more than a billion (10^9) elements within seconds on a cluster with 32 nodes using less than 10GB of memory per node.

cs.CG↗

arXiv2013

Approximate Cech Complexes in Low and High Dimensions

Čech complexes reveal valuable topological information about point sets at a certain scale in arbitrary dimensions, but the sheer size of these complexes limits their practical impact. While recent work introduced approximation techniques for filtrations of (Vietoris-)Rips complexes, a coarser version of Čech complexes, we propose the approximation of Čech filtrations directly. For fixed dimensional point set $S$, we present an approximation of the Čech filtration of $S$ by a sequence of complexes of size linear in the number of points. We generalize well-separated pair decompositions (WSPD) to well-separated simplicial decomposition (WSSD) in which every simplex defined on $S$ is covered by some element of WSSD. We give an efficient algorithm to compute a linear-sized WSSD in fixed dimensional spaces. Using a WSSD, we then present a linear-sized approximation of the filtration of Čech complex of $S$. We also present a generalization of the known fact that the Rips complex approximates the Čech complex by a factor of $\sqrt{2}$. We define a class of complexes that interpolate between Čech and Rips complexes and that, given any parameter $\eps > 0$, approximate the Čech complex by a factor $(1+\eps)$. Our complex can be represented by roughly $O(n^{\lceil 1/2\eps\rceil})$ simplices without any hidden dependence on the ambient dimension of the point set. Our results are based on an interesting link between Čech complex and coresets for minimum enclosing ball of high-dimensional point sets. As a consequence of our analysis, we show improved bounds on coresets that approximate the radius of the minimum enclosing ball.

cs.CG↗

arXiv2013

Root Refinement for Real Polynomials

We consider the problem of approximating all real roots of a square-free polynomial $f$. Given isolating intervals, our algorithm refines each of them to a width of $2^{-L}$ or less, that is, each of the roots is approximated to $L$ bits after the binary point. Our method provides a certified answer for arbitrary real polynomials, only considering finite approximations of the polynomial coefficients and choosing a suitable working precision adaptively. In this way, we get a correct algorithm that is simple to implement and practically efficient. Our algorithm uses the quadratic interval refinement method; we adapt that method to be able to cope with inaccuracies when evaluating $f$, without sacrificing its quadratic convergence behavior. We prove a bound on the bit complexity of our algorithm in terms of the degree of the polynomial, the size and the separation of the roots, that is, parameters exclusively related to the geometric location of the roots. Our bound is near optimal and significantly improves previous work on integer polynomials. Furthermore, it essentially matches the best known theoretical bounds on root approximation which are obtained by very sophisticated algorithms. We also investigate the practical behavior of the algorithm and demonstrate how closely the practical performance matches our asymptotic bounds.

cs.SC↗

arXiv2013

Embedding the dual complex of hyper-rectangular partitions

A rectangular partition is the partition of an (axis-aligned) rectangle into interior-disjoint rectangles. We ask whether a rectangular partition permits a "nice" drawing of its dual, that is, a straight-line embedding of it such that each dual vertex is placed into the rectangle that it represents. We show that deciding whether such a drawing exists is NP-complete. Moreover, we consider the drawing where a vertex is placed in the center of the represented rectangle and consider sufficient conditions for this drawing to be nice. This question is studied both in the plane and for the higher-dimensional generalization of rectangular partitions.

cs.CG↗

arXiv2013

Clear and Compress: Computing Persistent Homology in Chunks

We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then simplifying the unpaired columns, and finally applying standard reduction on the simplified matrix. The approach generalizes a technique by Günther et al., which uses discrete Morse Theory to compute persistence; we derive the same worst-case complexity bound in a more general context. The algorithm employs several practical optimization techniques which are of independent interest. Our sequential implementation of the algorithm is competitive with state-of-the-art methods, and we improve the performance through parallelized computation.

math.AT↗

arXiv2012

The Medusa of Spatial Sorting: 3D Kinetic Alpha Complexes and Implementation

Motivated by an application in cell biology, we consider spatial sorting processes defined by particles moving from an initial to a final configuration. We describe an algorithm for constructing a cell complex in space-time, called the medusa, that measures topological properties of the sorting process. The algorithm requires an extension of the kinetic data structures framework from Delaunay triangulations to fixed-radius alpha complexes. We report on several techniques to accelerate the computation.

cs.CG↗

arXiv2012

The Medusa of Spatial Sorting: Topological Construction

We consider the simultaneous movement of finitely many colored points in space, calling it a spatial sorting process. The name suggests a purpose that drives the collection to a configuration of increased or decreased order. Mapping such a process to a subset of space-time, we use persistent homology measurements of the time function to characterize the process topologically.

cs.CG↗

arXiv2011

Alexander Duality for Functions: the Persistent Behavior of Land and Water and Shore

This note contributes to the point calculus of persistent homology by extending Alexander duality to real-valued functions. Given a perfect Morse function $f: S^{n+1} \to [0,1]$ and a decomposition $S^{n+1} = U \cup V$ such that $M = \U \cap V$ is an $n$-manifold, we prove elementary relationships between the persistence diagrams of $f$ restricted to $U$, to $V$, and to $M$.

math.AT↗

arXiv2011

Deconstructing Approximate Offsets

We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using CGAL, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter δ; its running time additionally depends on δ. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.

cs.CG↗