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Horoball Hulls and Extents in Positive Definite Space

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 6844)

Abstract

The space of positive definite matrices P(n) is a Riemannian manifold with variable nonpositive curvature. It includes Euclidean space and hyperbolic space as submanifolds, and poses significant challenges for the design of algorithms for data analysis. In this paper, we develop foundational geometric structures and algorithms for analyzing collections of such matrices. A key technical contribution of this work is the use of horoballs, a natural generalization of halfspaces for non-positively curved Riemannian manifolds. We propose generalizations of the notion of a convex hull and a centerpoint and approximations of these structures using horoballs and based on novel decompositions of P(n). This leads to an algorithm for approximate hulls using a generalization of extents.

Keywords

  • Riemannian Manifold
  • Euclidean Space
  • Convex Hull
  • Voronoi Diagram
  • Hyperbolic Space

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This research was supported in part by NSF awards SGER-0841185 and CCF-0953066 and a subaward to the University of Utah under NSF award 0937060 to CRA.

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Fletcher, P.T., Moeller, J., Phillips, J.M., Venkatasubramanian, S. (2011). Horoball Hulls and Extents in Positive Definite Space. In: Dehne, F., Iacono, J., Sack, JR. (eds) Algorithms and Data Structures. WADS 2011. Lecture Notes in Computer Science, vol 6844. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22300-6_33

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  • DOI: https://doi.org/10.1007/978-3-642-22300-6_33

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22299-3

  • Online ISBN: 978-3-642-22300-6

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