Finding convex hull vertices in metric space

The convex hull has been extensively studied in computational geometry and its applications have spread over an impressive number of fields. How to find the convex hull is an important and challenging problem. Although many algorithms had been proposed for that, most of them can only tackle the problem in two or three dimensions and the biggest issue is that those algorithms rely on the samples' coordinates to find the convex hull. In this paper, we propose an approximation algorithm named FVDM, which only utilizes the information of the samples' distance matrix to find the convex hull. Experiments demonstrate that FVDM can effectively identify the vertices of the convex hull.

[1]  Timothy M. Chan Optimal output-sensitive convex hull algorithms in two and three dimensions , 1996, Discret. Comput. Geom..

[2]  Chih-Jen Lin,et al.  A Practical Guide to Support Vector Classication , 2008 .

[3]  Kristin P. Bennett,et al.  Duality and Geometry in SVM Classifiers , 2000, ICML.

[4]  Antoine Bordes,et al.  The Huller: A Simple and Efficient Online SVM , 2005, ECML.

[5]  Jonathan Foote,et al.  Summarizing video using non-negative similarity matrix factorization , 2002, 2002 IEEE Workshop on Multimedia Signal Processing..

[6]  David P. Dobkin,et al.  The quickhull algorithm for convex hulls , 1996, TOMS.

[7]  Pat Morin,et al.  Space-efficient planar convex hull algorithms , 2004, Theor. Comput. Sci..

[8]  F. P. Preparata,et al.  Convex hulls of finite sets of points in two and three dimensions , 1977, CACM.

[9]  Sergios Theodoridis,et al.  A geometric approach to Support Vector Machine (SVM) classification , 2006, IEEE Transactions on Neural Networks.

[10]  Ray A. Jarvis,et al.  On the Identification of the Convex Hull of a Finite Set of Points in the Plane , 1973, Inf. Process. Lett..

[11]  Bernard Haasdonk,et al.  Feature space interpretation of SVMs with indefinite kernels , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[12]  Nicolas Halbwachs,et al.  Delay Analysis in Synchronous Programs , 1993, CAV.

[13]  H. Kiers,et al.  Selecting among three-mode principal component models of different types and complexities: a numerical convex hull based method. , 2006, The British journal of mathematical and statistical psychology.

[14]  Ronald L. Graham,et al.  An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set , 1972, Inf. Process. Lett..

[15]  Heng Tao Shen,et al.  Principal Component Analysis , 2009, Encyclopedia of Biometrics.

[16]  Min Wang,et al.  Online Support Vector Machine Based on Convex Hull Vertices Selection , 2013, IEEE Transactions on Neural Networks and Learning Systems.

[17]  Yifei Wang,et al.  Geometric Algorithms to Large Margin Classifier Based on Affine Hulls , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[18]  A. Ardeshir Goshtasby,et al.  Point pattern matching using convex hull edges , 1985, IEEE Transactions on Systems, Man, and Cybernetics.

[19]  Donald R. Chand,et al.  An Algorithm for Convex Polytopes , 1970, JACM.

[20]  Pedro M. Ferreira,et al.  A simple algorithm for convex hull determination in high dimensions , 2013, 2013 IEEE 8th International Symposium on Intelligent Signal Processing.

[21]  Timothy M. Chan Output-sensitive results on convex hulls, extreme points, and related problems , 1995, SCG '95.