2,718 research outputs found
Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension
We study the combinatorial complexity of D-dimensional polyhedra defined as
the intersection of n halfspaces, with the property that the highest dimension
of any bounded face is much smaller than D. We show that, if d is the maximum
dimension of a bounded face, then the number of vertices of the polyhedron is
O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For
inputs in general position the number of bounded faces is O(n^d). For any fixed
d, we show how to compute the set of all vertices, how to determine the maximum
dimension of a bounded face of the polyhedron, and how to compute the set of
bounded faces in polynomial time, by solving a polynomial number of linear
programs
Colourful Simplicial Depth
Inspired by Barany's colourful Caratheodory theorem, we introduce a colourful
generalization of Liu's simplicial depth. We prove a parity property and
conjecture that the minimum colourful simplicial depth of any core point in any
d-dimensional configuration is d^2+1 and that the maximum is d^(d+1)+1. We
exhibit configurations attaining each of these depths and apply our results to
the problem of bounding monochrome (non-colourful) simplicial depth.Comment: 18 pages, 5 figues. Minor polishin
Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes
Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational
polytope has a quasi-period less than the denominator of that polytope. This
phenomenon is poorly understood, and all known cases in which it occurs have
been proven with ad hoc methods. In this note, we present a conjectural
explanation for quasi-period collapse in rational polytopes. We show that this
explanation applies to some previous cases appearing in the literature. We also
exhibit examples of Ehrhart polynomials of rational polytopes that are not the
Ehrhart polynomials of any integral polytope.
Our approach depends on the invariance of the Ehrhart quasi-polynomial under
the action of affine unimodular transformations. Motivated by the similarity of
this idea to the scissors congruence problem, we explore the development of a
Dehn-like invariant for rational polytopes in the lattice setting.Comment: 8 pages, 3 figures, to appear in the proceedings of Integer points in
polyhedra, June 11 -- June 15, 2006, Snowbird, U
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
Asymptotically efficient triangulations of the d-cube
Let and be polytopes, the first of "low" dimension and the second of
"high" dimension. We show how to triangulate the product
efficiently (i.e., with few simplices) starting with a given triangulation of
. Our method has a computational part, where we need to compute an efficient
triangulation of , for a (small) natural number of our
choice. denotes the -simplex.
Our procedure can be applied to obtain (asymptotically) efficient
triangulations of the cube : We decompose , for
a small . Then we recursively assume we have obtained an efficient
triangulation of the second factor and use our method to triangulate the
product. The outcome is that using and , we can triangulate
with simplices, instead of the achievable
before.Comment: 19 pages, 6 figures. Only minor changes from previous versions, some
suggested by anonymous referees. Paper accepted in "Discrete and
Computational Geometry
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