307 research outputs found
Polyhedral Cones of Magic Cubes and Squares
Using computational algebraic geometry techniques and Hilbert bases of
polyhedral cones we derive explicit formulas and generating functions for the
number of magic squares and magic cubes.Comment: 14 page
On f-vectors of Minkowski additions of convex polytopes
The objective of this paper is to present two types of results on Minkowski
sums of convex polytopes. The first is about a special class of polytopes we
call perfectly centered and the combinatorial properties of the Minkowski sum
with their own dual. In particular, we have a characterization of face lattice
of the sum in terms of the face lattice of a given perfectly centered polytope.
Exact face counting formulas are then obtained for perfectly centered simplices
and hypercubes. The second type of results concerns tight upper bounds for the
f-vectors of Minkowski sums of several polytopes.Comment: 13 pages, submitted to Discrete & Computational Geometr
Compressed sensing of data with a known distribution
Compressed sensing is a technique for recovering an unknown sparse signal
from a small number of linear measurements. When the measurement matrix is
random, the number of measurements required for perfect recovery exhibits a
phase transition: there is a threshold on the number of measurements after
which the probability of exact recovery quickly goes from very small to very
large. In this work we are able to reduce this threshold by incorporating
statistical information about the data we wish to recover. Our algorithm works
by minimizing a suitably weighted -norm, where the weights are chosen
so that the expected statistical dimension of the corresponding descent cone is
minimized. We also provide new discrete-geometry-based Monte Carlo algorithms
for computing intrinsic volumes of such descent cones, allowing us to bound the
failure probability of our methods.Comment: 22 pages, 7 figures. New colorblind safe figures. Sections 3 and 4
completely rewritten. Minor typos fixe
Space Exploration via Proximity Search
We investigate what computational tasks can be performed on a point set in
, if we are only given black-box access to it via nearest-neighbor
search. This is a reasonable assumption if the underlying point set is either
provided implicitly, or it is stored in a data structure that can answer such
queries. In particular, we show the following: (A) One can compute an
approximate bi-criteria -center clustering of the point set, and more
generally compute a greedy permutation of the point set. (B) One can decide if
a query point is (approximately) inside the convex-hull of the point set.
We also investigate the problem of clustering the given point set, such that
meaningful proximity queries can be carried out on the centers of the clusters,
instead of the whole point set
From Steiner Formulas for Cones to Concentration of Intrinsic Volumes
The intrinsic volumes of a convex cone are geometric functionals that return
basic structural information about the cone. Recent research has demonstrated
that conic intrinsic volumes are valuable for understanding the behavior of
random convex optimization problems. This paper develops a systematic technique
for studying conic intrinsic volumes using methods from probability. At the
heart of this approach is a general Steiner formula for cones. This result
converts questions about the intrinsic volumes into questions about the
projection of a Gaussian random vector onto the cone, which can then be
resolved using tools from Gaussian analysis. The approach leads to new
identities and bounds for the intrinsic volumes of a cone, including a
near-optimal concentration inequality.Comment: This version corrects errors in Propositions 3.3 and 3.4 and in Lemma
8.3 that appear in the published versio
f-Vectors of Minkowski Additions of Convex Polytopes
The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of the face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face counting formulas are then obtained for perfectly centered simplices and hypercubes. The second type of results concerns tight upper bounds for the f-vectors of Minkowski sums of several polytope
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