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

\v{C}ech-Delaunay gradient flow and homology inference for self-maps

We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspace of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for \v{C}ech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive \v{C}ech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems.Comment: 22 pages, 8 figure

Multi-view 3D retrieval using silhouette intersection and multi-scale contour representation

We describe in this paper two methods for 3D shape indexing and retrieval that we apply on two data collections of the SHREC - SHape Retrieval Contest 2007: Watertight models and 3D CAD models. Both methods are based on a set of 2D multi-views after a pose and scale normalization of the models using PCA and the enclosing sphere. In all views we extract the models silhouettes and compare them pairwise. In the first method the similitude measure is obtained by integrating on the pairs of views the difference between the areas of the silhouettes union and the silhouettes intersection. In the second method we consider the external contour of the silhouettes, extract their convexities and concavities at different scale levels and build a multiscale representation. The pairs of contours are then compared by elastic matching achieved by using dynamic programming. Comparisons of the two methods are shown with their respective strengths and weaknesses

Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms

We present a technical survey on the state of the art approaches in data reduction and the coreset framework. These include geometric decompositions, gradient methods, random sampling, sketching and random projections. We further outline their importance for the design of streaming algorithms and give a brief overview on lower bounding techniques