On approximating the Riemannian 1-center

We generalize the Euclidean 1-center approximation algorithm of Badoiu and Clarkson (2003) [6] to arbitrary Riemannian geometries, and study the corresponding convergence rate. We then show how to instantiate this generic algorithm to two particular settings: (1) the hyperbolic geometry, and (2) the Riemannian manifold of symmetric positive definite matrices.

[1]  Huibo Ji,et al.  Optimization approaches on smooth manifolds , 2007 .

[2]  S. Lang.  Fundamentals of differential geometry , 1998 .

[3]  J. Ratcliffe.  Foundations of Hyperbolic Manifolds , 2019, Graduate Texts in Mathematics.

[4]  Silvere Bonnabel,et al.  Stochastic Gradient Descent on Riemannian Manifolds , 2011, IEEE Transactions on Automatic Control.

[5]  M. Arnaudon,et al.  Stochastic algorithms for computing means of probability measures , 2011, 1106.5106.

[6]  J. Cheeger,et al.  Comparison theorems in Riemannian geometry , 1975 .

[7]  Rama Chellappa,et al.  Statistical Computations on Grassmann and Stiefel Manifolds for Image and Video-Based Recognition , 2011, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[8]  Hermann A. Maurer,et al.  New Results and New Trends in Computer Science , 1991, Lecture Notes in Computer Science.

[9]  Frank Nielsen,et al.  Fitting the Smallest Enclosing Bregman Ball , 2005, ECML.

[10]  Kenneth L. Clarkson,et al.  Optimal core-sets for balls , 2008, Comput. Geom..

[11]  Frank Nielsen,et al.  Hyperbolic Voronoi Diagrams Made Easy , 2009, 2010 International Conference on Computational Science and Its Applications.

[12]  Frank Nielsen,et al.  Approximating Smallest Enclosing Balls with Applications to Machine Learning , 2009, Int. J. Comput. Geom. Appl..

[13]  Emo Welzl,et al.  Smallest enclosing disks (balls and ellipsoids) , 1991, New Results and New Trends in Computer Science.

[14]  Kenneth L. Clarkson,et al.  Smaller core-sets for balls , 2003, SODA '03.

[15]  Le Yang.  Riemannian Median and Its Estimation , 2009, 0911.3474.

[16]  Xavier Pennec,et al.  Statistical Computing on Manifolds: From Riemannian Geometry to Computational Anatomy , 2009, ETVC.

[17]  B. Afsari.  MEANS AND AVERAGING ON RIEMANNIAN MANIFOLDS , 2009 .

[18]  D. Bertsekas,et al.  Convergen e Rate of In remental Subgradient Algorithms , 2000 .

[19]  B. Afsari.  Riemannian Lp center of mass: existence, uniqueness, and convexity , 2011 .

[20]  I. Chavel.  Riemannian Geometry: Subject Index , 2006 .

[21]  N. Hitchin.  A panoramic view of riemannian geometry , 2006 .