4,791 research outputs found

    Quasiconvex Programming

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    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

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    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

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    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 (d≠4d \ne 4) 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

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    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 inputsĂ—outputs\mathrm{inputs}\times\mathrm{outputs} 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 inputsĂ—outputs\mathrm{inputs}\times\mathrm{outputs} 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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