Abstract
Geometric interpretations of Support Vector Machines (SVMs) have introduced the concept of a reduced convex hull. A reduced convex hull is the set of all convex combinations of a set of points where the weight any single point can be assigned is bounded from above by a constant. This paper decouples reduced convex hulls from their origins in SVMs and allows them to be constructed independently. Two algorithms for the computation of reduced convex hulls are presented – a simple recursive algorithm for points in the plane and an algorithm for points in an arbitrary dimensional space. Upper bounds on the number of vertices and facets in a reduced convex hull are used to analyze the worst-case complexity of the algorithms.
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References
Bennett, K.P., Bredensteiner, E.J.: Duality and geometry in SVM classifiers. In: ICML 2000: Proceedings of the Seventeenth International Conference on Machine Learning, pp. 57–64. Morgan Kaufmann Publishers Inc., San Francisco (2000)
Crisp, D.J., Burges, C.J.C.: A geometric interpretation of ν-SVM classifiers. In: Solla, S.A., Leen, T.K., Müller, K.R. (eds.) Advances in Neural Information Processing Systems 12, Papers from Neural Information Processing Systems (NIPS), Denver, CO, USA, pp. 244–251. MIT Press, Cambridge (1999)
Bern, M., Eppstein, D.: Optimization over zonotopes and training Support Vector Machines. In: Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 2001. LNCS, vol. 2125, pp. 111–121. Springer, Heidelberg (2001)
Mavroforakis, M.E., Sdralis, M., Theodoridis, S.: A novel SVM geometric algorithm based on reduced convex hulls. In: 18th International Conference on Pattern Recognition, vol. 2, pp. 564–568 (2006)
Mavroforakis, M.E., Theodoridis, S.: A geometric approach to Support Vector Machine (SVM) classification. IEEE Transactions on Neural Networks 17(3), 671–682 (2006)
Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software 22(4), 469–483 (1996)
Aurenhammer, F.: Voronoi diagrams – a survey of a fundamental geometric data structure. ACM Computing Surveys 23(3), 345–405 (1991)
Preparata, F.P., Shamos, M.I.: Computational Geometry, 2nd edn. Springer, New York (1988)
Eddy, W.F.: A new convex hull algorithm for planar sets. ACM Transactions on Mathematical Software 3(4), 398–403 (1977)
Bykat, A.: Convex hull of a finite set of points in two dimensions. Information Processing Letters 7, 296–298 (1978)
Green, P.J., Silverman, B.W.: Constructing the convex hull of a set of points in the plane. The Computer Journal 22(3), 262–266 (1978)
Kallay, M.: Convex hull algorithms in higher dimensions. Department of Mathematics, University of Oklahoma (1981) (unpublished manuscript)
Klee, V.: Convex polytopes and linear programming. In: IBM Scientific Computing Symposium: Combinatorial Problems, pp. 123–158. IBM, Armonk (1966)
O’Rourke, J.: Computation Geometry in C, 2nd edn. Cambridge University Press, Cambridge (1998)
Clarkson, K.L., Mehlhorn, K., Seidel, R.: Four results on randomized incremental constructions. Computational Geometry: Theory and Applications 3(4), 185–212 (1993)
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Goodrich, B., Albrecht, D., Tischer, P. (2009). Algorithms for the Computation of Reduced Convex Hulls. In: Nicholson, A., Li, X. (eds) AI 2009: Advances in Artificial Intelligence. AI 2009. Lecture Notes in Computer Science(), vol 5866. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10439-8_24
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DOI: https://doi.org/10.1007/978-3-642-10439-8_24
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