4,791 research outputs found
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
A Combinatorial Solution to Non-Rigid 3D Shape-to-Image Matching
We propose a combinatorial solution for the problem of non-rigidly matching a
3D shape to 3D image data. To this end, we model the shape as a triangular mesh
and allow each triangle of this mesh to be rigidly transformed to achieve a
suitable matching to the image. By penalising the distance and the relative
rotation between neighbouring triangles our matching compromises between image
and shape information. In this paper, we resolve two major challenges: Firstly,
we address the resulting large and NP-hard combinatorial problem with a
suitable graph-theoretic approach. Secondly, we propose an efficient
discretisation of the unbounded 6-dimensional Lie group SE(3). To our knowledge
this is the first combinatorial formulation for non-rigid 3D shape-to-image
matching. In contrast to existing local (gradient descent) optimisation
methods, we obtain solutions that do not require a good initialisation and that
are within a bound of the optimal solution. We evaluate the proposed method on
the two problems of non-rigid 3D shape-to-shape and non-rigid 3D shape-to-image
registration and demonstrate that it provides promising results.Comment: 10 pages, 7 figure
Collapsibility of CAT(0) spaces
Collapsibility is a combinatorial strengthening of contractibility. We relate
this property to metric geometry by proving the collapsibility of any complex
that is CAT(0) with a metric for which all vertex stars are convex. This
strengthens and generalizes a result by Crowley. Further consequences of our
work are:
(1) All CAT(0) cube complexes are collapsible.
(2) Any triangulated manifold admits a CAT(0) metric if and only if it admits
collapsible triangulations.
(3) All contractible d-manifolds () admit collapsible CAT(0)
triangulations. This discretizes a classical result by Ancel--Guilbault.Comment: 27 pages, 3 figures. The part on collapsibility of convex complexes
has been removed and forms a new paper, called "Barycentric subdivisions of
convexes complex are collapsible" (arXiv:1709.07930). The part on enumeration
of manifolds has also been removed and forms now a third paper, called "A
Cheeger-type exponential bound for the number of triangulated manifolds"
(arXiv:1710.00130
Asymptotic distribution of conical-hull estimators of directional edges
Nonparametric data envelopment analysis (DEA) estimators have been widely
applied in analysis of productive efficiency. Typically they are defined in
terms of convex-hulls of the observed combinations of
in a sample of enterprises. The shape
of the convex-hull relies on a hypothesis on the shape of the technology,
defined as the boundary of the set of technically attainable points in the
space. So far, only the statistical
properties of the smallest convex polyhedron enveloping the data points has
been considered which corresponds to a situation where the technology presents
variable returns-to-scale (VRS). This paper analyzes the case where the most
common constant returns-to-scale (CRS) hypothesis is assumed. Here the DEA is
defined as the smallest conical-hull with vertex at the origin enveloping the
cloud of observed points. In this paper we determine the asymptotic properties
of this estimator, showing that the rate of convergence is better than for the
VRS estimator. We derive also its asymptotic sampling distribution with a
practical way to simulate it. This allows to define a bias-corrected estimator
and to build confidence intervals for the frontier. We compare in a simulated
example the bias-corrected estimator with the original conical-hull estimator
and show its superiority in terms of median squared error.Comment: Published in at http://dx.doi.org/10.1214/09-AOS746 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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