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## Computing Hulls And Centerpoints In Positive Definite Space (2009)

Citations: | 1 - 0 self |

### Citations

1222 |
Kernel Methods for Pattern Analysis
- Shawe-Taylor, Cristianini
- 2004
(Show Context)
Citation Context ...in P(3) model the flow of water at each voxel of a brain scan. In mechanical engineering [11], stress tensors are modeled as elements of P(6). Kernel matrices in machine learning are elements of P(n) =-=[25]-=-. In all these areas, a problem of great interest is the analysis [13, 14] of collections of such matrices (finding central points, clustering, doing regression). For all of these problems, we need th... |

704 |
Haefliger,Metric Spaces of Non-Positive Curvature
- Bridson, A
- 1999
(Show Context)
Citation Context ...oint, but is guaranteed to be close to such a point. 1.2 Related Work The mathematics of Riemannian manifolds, Cartan-Hadamard manifolds and P(n) is well-understood: the book by Bridson and Haefliger =-=[6]-=- is an invaluable reference on metric spaces of nonpositive curvature, and Bhatia [5] provides a detailed study of P(n) in particular. However, there are many fewer algorithmic results for problems in... |

361 | Mr diffusion tensor spectroscopy and imaging
- Basser, Mattiello, et al.
- 1994
(Show Context)
Citation Context ...re are many application areas where the basic objects of interest, rather than points in Euclidean space, are symmetric positive-definite n × n matrices (denoted by P(n)). In diffusion tensor imaging =-=[3]-=-, matrices in P(3) model the flow of water at each voxel of a brain scan. In mechanical engineering [11], stress tensors are modeled as elements of P(6). Kernel matrices in machine learning are elemen... |

181 |
A Panoramic View of Riemannian Geometry
- Berger
(Show Context)
Citation Context ...sense) of the input points. A significant obstacle to the convex hull in P(n) is that it is not even known whether the convex hull of a finite collection of points in P(n) can be represented finitely =-=[4]-=-. Another approach to defining the convex hull is via halfspaces: we can define the convex hull in Euclidean space as the intersection of all halfspaces that contain all the points. Unfortunately, eve... |

180 |
Positive Definite Matrices
- Bhatia
- 2007
(Show Context)
Citation Context ... of Riemannian manifolds, Cartan-Hadamard manifolds and P(n) is well-understood: the book by Bridson and Haefliger [6] is an invaluable reference on metric spaces of nonpositive curvature, and Bhatia =-=[5]-=- provides a detailed study of P(n) in particular. However, there are many fewer algorithmic results for problems in these spaces. To the best of our knowledge, the only prior work on algorithms for po... |

137 |
The ordering of multivariate data
- Barnett
- 1976
(Show Context)
Citation Context ... computational geometry, the convex hull also provides a compact description of the boundary of a data set, can be used to define the center of a data set (via the notion of convex hull peeling depth =-=[23, 2]-=-), and also captures extremal properties of a data set like its diameter, width and bounding volume (even in its approximate form [1]). The convex hull of a set of points in P(n) can be naturally defi... |

123 | Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors
- Fletcher, Joshi
- 2004
(Show Context)
Citation Context ...anical engineering [11], stress tensors are modeled as elements of P(6). Kernel matrices in machine learning are elements of P(n) [25]. In all these areas, a problem of great interest is the analysis =-=[13, 14]-=- of collections of such matrices (finding central points, clustering, doing regression). For all of these problems, we need the same kinds of geometric tools available to us in Euclidean space, includ... |

120 |
Lectures in Discrete Geometry
- Matouˇsek
- 2002
(Show Context)
Citation Context ...ossible center point p there is a horoball that contains at most 1 point of X. Hence, there can be no center point. 12Horo-center points in P(n). The “simple” proof of the existence of center points =-=[19]-=- uses Helly’s theorem to show that a horo-center point always exist, and then in Euclidean space a halfspace separation theorem can be used to show that a horo-center point is also a center point. We ... |

117 | Approximating extent measures of points
- Agarwal, Har-Peled, et al.
- 2004
(Show Context)
Citation Context ...ter of a data set (via the notion of convex hull peeling depth [23, 2]), and also captures extremal properties of a data set like its diameter, width and bounding volume (even in its approximate form =-=[1]-=-). The convex hull of a set of points in P(n) can be naturally defined as the intersection of all convex sets containing the points. Alternatively, it can be defined as the set of all points that are ... |

92 | On linear-time deterministic algorithms for optimization problems in fixed dimension
- Chazelle, Matouˇsek
- 1996
(Show Context)
Citation Context ...rdinality of the largest basis is the combinatorial dimension of the LP-type problem. LP-type problems with constant combinatorial dimensions can be solved in time linear in the number of constraints =-=[7]-=-. Lemma 5.1. A set H of horoballs in P(n), and a function ω(G) = min T p∈ det(p) is an LP-type problem with H constant combinatorial dimension. Proof. Monotonicity holds since in adding more horoballs... |

91 |
A combinatorial bound for linear programming and related problems
- Sharir, Welzl
- 1992
(Show Context)
Citation Context ...nstructing horocenter points in P(n) and then provide an algorithm for an approximate horocenter point in P(n). Before we begin we need a useful definition of a family of problems. An LP-type Problem =-=[24]-=- takes as input a set of constraints H and a function ω : 2 H → R ¯ that we seek to minimize, and it has the following two properties. MONOTONICITY: For any F ⊆ G ⊆ H, ω(F ) ≤ ω(G). LOCALITY: For any ... |

72 | Delaunay Triangulations and Voronoi Diagrams for Riemannian Manifolds
- Leibon, Letscher
- 2000
(Show Context)
Citation Context ...h more tractable. The Poincaré and Klein models of hyperbolic space preserve different properties of Euclidean space, and many algorithm carry over directly with no modifications. Leibon and Letscher =-=[18]-=- were the first to study basic geometric primitives in general Riemannian manifolds, constructing Voronoi diagrams and Delaunay triangulations for sufficiently dense point sets in these spaces. Eppste... |

51 | Approximating center points with iterative radon points
- Clarkson, Eppstein, et al.
- 1996
(Show Context)
Citation Context ...t any halfspace that contains p ¯ also contains at least N/(d + 1) points from X. Center points always exist [22] and there exists several algorithms for computing them exactly [15] and approximately =-=[10, 20]-=-. We cannot directly replicate the notion of center points in P(n) with horoballs. Instead we replace it with a slightly d weaker notion, which is equivalent in Euclidean space. A horo-center point p ... |

37 | The geometric median on Riemannian manifolds with application to robust atlas estimation
- Fletcher, Venkatasubramanian, et al.
(Show Context)
Citation Context ...anical engineering [11], stress tensors are modeled as elements of P(6). Kernel matrices in machine learning are elements of P(n) [25]. In all these areas, a problem of great interest is the analysis =-=[13, 14]-=- of collections of such matrices (finding central points, clustering, doing regression). For all of these problems, we need the same kinds of geometric tools available to us in Euclidean space, includ... |

33 |
Computing a centerpoint of a finite planar set of points in linear time. Discrete and Computational Geometry 12
- Jadhav, Mukhopadhyay
- 1994
(Show Context)
Citation Context ... N has the property that any halfspace that contains p ¯ also contains at least N/(d + 1) points from X. Center points always exist [22] and there exists several algorithms for computing them exactly =-=[15]-=- and approximately [10, 20]. We cannot directly replicate the notion of center points in P(n) with horoballs. Instead we replace it with a slightly d weaker notion, which is equivalent in Euclidean sp... |

31 |
A theorem on general measure
- Rado
- 1947
(Show Context)
Citation Context ... Points d In Euclidean space a center point p of a set X ⊂ R of size N has the property that any halfspace that contains p ¯ also contains at least N/(d + 1) points from X. Center points always exist =-=[22]-=- and there exists several algorithms for computing them exactly [15] and approximately [10, 20]. We cannot directly replicate the notion of center points in P(n) with horoballs. Instead we replace it ... |

17 | Geometry and Statistics: Problems at the Interface
- SHAMOS
- 1976
(Show Context)
Citation Context ... computational geometry, the convex hull also provides a compact description of the boundary of a data set, can be used to define the center of a data set (via the notion of convex hull peeling depth =-=[23, 2]-=-), and also captures extremal properties of a data set like its diameter, width and bounding volume (even in its approximate form [1]). The convex hull of a set of points in P(n) can be naturally defi... |

12 |
Symmetric positive-definite matrices: From geometry to applications and visualization
- Moakher
- 2006
(Show Context)
Citation Context ...ar. However, there are many fewer algorithmic results for problems in these spaces. To the best of our knowledge, the only prior work on algorithms for positive definite space are the work by Moakher =-=[21]-=- on mean shapes in positive definite space, and papers by Fletcher and Joshi [13] on doing principal geodesic analysis in symmetric spaces, and the robust median algorithms of Fletcher et al [14] for ... |

10 | Packing and covering δ-hyperbolic spaces by balls
- Chepoi, Estellon
(Show Context)
Citation Context ...for points in δ-hyperbolic space; these spaces are a combinatorial generalization of negatively curved space and are characterized by global, rather than local, definitions of curvature. Chepoi et al =-=[8, 9]-=- advanced this line of research, providing algorithms for computing the diameter and minimum enclosing ball of collections of points in δ-hyperbolic space. 2 Preliminaries P(n) is the set of symmetric... |

9 |
Squarepants in a tree: sum of subtree clustering and hyperbolic pants decomposition
- Eppstein
(Show Context)
Citation Context ...re the first to study basic geometric primitives in general Riemannian manifolds, constructing Voronoi diagrams and Delaunay triangulations for sufficiently dense point sets in these spaces. Eppstein =-=[12]-=- described hierarchical clustering algorithms in hyperbolic space. Krauthgamer and Lee [16] studied the nearest neighbor problem for points in δ-hyperbolic space; these spaces are a combinatorial gene... |

7 |
The structure of the linear anisotropic elastic symmetries
- Cowin, Mehrabadi
(Show Context)
Citation Context ...e, are symmetric positive-definite n × n matrices (denoted by P(n)). In diffusion tensor imaging [3], matrices in P(3) model the flow of water at each voxel of a brain scan. In mechanical engineering =-=[11]-=-, stress tensors are modeled as elements of P(6). Kernel matrices in machine learning are elements of P(n) [25]. In all these areas, a problem of great interest is the analysis [13, 14] of collections... |

4 |
Algorithms on negatively curved spaces, FOCS
- Krauthgamer, Lee
- 2006
(Show Context)
Citation Context ...ing Voronoi diagrams and Delaunay triangulations for sufficiently dense point sets in these spaces. Eppstein [12] described hierarchical clustering algorithms in hyperbolic space. Krauthgamer and Lee =-=[16]-=- studied the nearest neighbor problem for points in δ-hyperbolic space; these spaces are a combinatorial generalization of negatively curved space and are characterized by global, rather than local, d... |

3 |
approximating trees of deltahyperbolicgeodesic spaces and graphs
- Chepoi, Dragan, et al.
- 2008
(Show Context)
Citation Context ...for points in δ-hyperbolic space; these spaces are a combinatorial generalization of negatively curved space and are characterized by global, rather than local, definitions of curvature. Chepoi et al =-=[8, 9]-=- advanced this line of research, providing algorithms for computing the diameter and minimum enclosing ball of collections of points in δ-hyperbolic space. 2 Preliminaries P(n) is the set of symmetric... |

2 |
Helly’s intersection theorem on manifolds of nonpositive curvature
- Ledyaev, Treiman, et al.
(Show Context)
Citation Context ... properties when not defined in R . ¯ Theorem 5.2. Any set X ⊂ P(n) has a horo-center point. Proof. We use the following Helly Theorem on Cartan-Hadamard manifolds (which include P(n)) of dimension d =-=[17]-=-. For a family F of closed convex sets, if any set of d + 1 sets from F contain a common point, then the intersection of all sets in F contain a common point. In P(n) we consider the family F of close... |