2,127 research outputs found
Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics
Using the geodesic distance on the -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 , 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 -dimensional
Euclidean space. As proved by Antonelli and collaborators, an orthant section
of the -sphere is isometric to the standard -simplex equipped with the
Fisher information metric. It follows that the latter space has similar
stochastic properties as the -dimensional Euclidean space. Our results are
therefore relevant in information geometry and in population genetics
Weighted Poisson-Delaunay Mosaics
Slicing a Voronoi tessellation in with a -plane gives a
-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 -plane gives a generalized discrete Morse function. Assuming the
Voronoi tessellation is generated by a Poisson point process in ,
we study the expected number of simplices in the -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
The Minimum Expectation Selection Problem
We define the min-min expectation selection problem (resp. max-min
expectation selection problem) to be that of selecting k out of n given
discrete probability distributions, to minimize (resp. maximize) the
expectation of the minimum value resulting when independent random variables
are drawn from the selected distributions. We assume each distribution has
finitely many atoms. Let d be the number of distinct values in the support of
the distributions. We show that if d is a constant greater than 2, the min-min
expectation problem is NP-complete but admits a fully polynomial time
approximation scheme. For d an arbitrary integer, it is NP-hard to approximate
the min-min expectation problem with any constant approximation factor. The
max-min expectation problem is polynomially solvable for constant d; we leave
open its complexity for variable d. We also show similar results for binary
selection problems in which we must choose one distribution from each of n
pairs of distributions.Comment: 13 pages, 1 figure. Full version of paper presented at 10th Int.
Conf. Random Structures and Algorithms, Poznan, Poland, August 200
Three-dimensional alpha shapes
Frequently, data in scientific computing is in its abstract form a finite
point set in space, and it is sometimes useful or required to compute what one
might call the ``shape'' of the set. For that purpose, this paper introduces
the formal notion of the family of -shapes of a finite point set in
\Real^3. Each shape is a well-defined polytope, derived from the Delaunay
triangulation of the point set, with a parameter \alpha \in \Real controlling
the desired level of detail. An algorithm is presented that constructs the
entire family of shapes for a given set of size in time , worst
case. A robust implementation of the algorithm is discussed and several
applications in the area of scientific computing are mentioned.Comment: 32 page
Topological Data Analysis with Bregman Divergences
Given a finite set in a metric space, the topological analysis generalizes
hierarchical clustering using a 1-parameter family of homology groups to
quantify connectivity in all dimensions. The connectivity is compactly
described by the persistence diagram. One limitation of the current framework
is the reliance on metric distances, whereas in many practical applications
objects are compared by non-metric dissimilarity measures. Examples are the
Kullback-Leibler divergence, which is commonly used for comparing text and
images, and the Itakura-Saito divergence, popular for speech and sound. These
are two members of the broad family of dissimilarities called Bregman
divergences.
We show that the framework of topological data analysis can be extended to
general Bregman divergences, widening the scope of possible applications. In
particular, we prove that appropriately generalized Cech and Delaunay (alpha)
complexes capture the correct homotopy type, namely that of the corresponding
union of Bregman balls. Consequently, their filtrations give the correct
persistence diagram, namely the one generated by the uniformly growing Bregman
balls. Moreover, we show that unlike the metric setting, the filtration of
Vietoris-Rips complexes may fail to approximate the persistence diagram. We
propose algorithms to compute the thus generalized Cech, Vietoris-Rips and
Delaunay complexes and experimentally test their efficiency. Lastly, we explain
their surprisingly good performance by making a connection with discrete Morse
theory
The Morse theory of \v{C}ech and Delaunay complexes
Given a finite set of points in and a radius parameter, we
study the \v{C}ech, Delaunay-\v{C}ech, Delaunay (or Alpha), and Wrap complexes
in the light of generalized discrete Morse theory. Establishing the \v{C}ech
and Delaunay complexes as sublevel sets of generalized discrete Morse
functions, we prove that the four complexes are simple-homotopy equivalent by a
sequence of simplicial collapses, which are explicitly described by a single
discrete gradient field.Comment: 21 pages, 2 figures, improved expositio
Delaunay Hodge Star
We define signed dual volumes at all dimensions for circumcentric dual
meshes. We show that for pairwise Delaunay triangulations with mild boundary
assumptions these signed dual volumes are positive. This allows the use of such
Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge
star operator can now be correctly defined for such meshes. This operator is
crucial for DEC and is a diagonal matrix with the ratio of primal and dual
volumes along the diagonal. A correct definition requires that all entries be
positive. DEC is a framework for numerically solving differential equations on
meshes and for geometry processing tasks and has had considerable impact in
computer graphics and scientific computing. Our result allows the use of DEC
with a much larger class of meshes than was previously considered possible.Comment: Corrected error in Figure 1 (columns 3 and 4) and Figure 6 and a
formula error in Section 2. All mathematical statements (theorems and lemmas)
are unchanged. The previous arXiv version v3 (minus the Appendix) appeared in
the journal Computer-Aided Desig
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 G\"unther 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.Comment: This result was presented at TopoInVis 2013
(http://www.sci.utah.edu/topoinvis13.html
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