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A proximal technique for computing the Karcher mean of symmetric positive definite matrices

Ronaldo Malheiros Gregório(rgregor***at***ufrrj.br)
Paulo Roberto Oliveira(poliveir***at***cos.ufrj)

Abstract: This paper presents a proximal point approach for computing the Riemannian or intrinsic Karcher mean of symmetric positive definite matrices. Our method derives from proximal point algorithm with Schur decomposition developed to compute minimum points of convex functions on symmetric positive definite matrices set when it is seen as a Hadamard manifold. The main idea of the original method is preserved. However, here, orthogonal matrices are updated as solutions of optimization subproblems on orthogonal group. Hence, the proximal trajectory is built on solving iteratively Riemannian optimization subproblems alternately on diagonal positive definite matrices set and orthogonal group. No Cholesky factorization is made over variables or datum of the problem. Bunches of numerical experiments, for n=2, . . ., 10, and illustrations of the computational behavior of Riemannian gradient descent, proximal point and Richardson-like algorithms are presented at the end.

Keywords: Karcher mean, symmetric positive definite matrices, proximal point algorithm, geodesic convexity, orthogonal group

Category 1: Convex and Nonsmooth Optimization (Generalized Convexity/Monoticity )

Category 2: Global Optimization

Citation: This work has been developed by researchers of Departamento de Tecnologias e Linguagens of Universidade Federal Rural do Rio de Janeiro and Programa de Engenharia de Sistemas e Computação of Universidade Federal do Rio de Janeiro and it has been supported by Faperj as part of the research project intituled "Algoritmo de ponto proximal com decomposições de Schur em domínios de positividade", set in 2012, March.

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Entry Submitted: 05/20/2013
Entry Accepted: 05/20/2013
Entry Last Modified: 05/20/2013

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