Delaunay triangulations and Voronoi diagrams for Riemannian manifolds
Proceedings of the sixteenth annual symposium on Computational geometry
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Advances in Applied Mathematics
Kernel Methods for Pattern Analysis
Kernel Methods for Pattern Analysis
Semi-Supervised Learning on Riemannian Manifolds
Machine Learning
Approximating extent measures of points
Journal of the ACM (JACM)
A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices
SIAM Journal on Matrix Analysis and Applications
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International Journal of Computer Vision
Algorithms on negatively curved spaces
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs
Proceedings of the twenty-fourth annual symposium on Computational geometry
Packing and Covering δ-Hyperbolic Spaces by Balls
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
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ACM Transactions on Algorithms (TALG)
Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank
SIAM Journal on Matrix Analysis and Applications
Surface sampling and the intrinsic Voronoi diagram
SGP '08 Proceedings of the Symposium on Geometry Processing
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IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
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IEEE Transactions on Signal Processing
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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.