197637
20181203023439.0
10689613
000311840500022
ISI
ARTICLE
A survey and comparison of contemporary algorithms for computing the matrix geometric mean
2012
2012
Journal Articles
In this paper we present a survey of various algorithms for computing matrix geometric means and derive new second-order optimization algorithms to compute the Karcher mean. These new algorithms are constructed using the standard definition of the Riemannian Hessian. The survey includes the ALM list of desired properties for a geometric mean, the analytical expression for the mean of two matrices, algorithms based on the centroid computation in Euclidean (flat) space, and Riemannian optimization techniques to compute the Karcher mean (preceded by a short introduction into differential geometry). A change of metric is considered in the optimization techniques to reduce the complexity of the structures used in these algorithms. Numerical experiments are presented to compare the existing and the newly developed algorithms. We conclude that currently first-order algorithms are best suited for this optimization problem as the size and/or number of the matrices increase. Copyright © 2012, Kent State University.
matrix geometric mean
positive definite matrices
Karcher mean
Riemannian optimization
Jeuris, B.
Vandebril, R.
Vandereycken, B.
379-402
Electronic Transactions on Numerical Analysis
39
n/a
408864
n/a
http://infoscience.epfl.ch/record/197637/files/pp379-402.pdf
ANCHP
252494
U12478
oai:infoscience.tind.io:197637
article
SB
GLOBAL_SET
213191
148230
EPFL-ARTICLE-197637
EPFL
PUBLISHED
REVIEWED
ARTICLE