000197637 001__ 197637
000197637 005__ 20181114202503.0
000197637 022__ $$a10689613
000197637 02470 $$2ISI$$a000311840500022
000197637 037__ $$aARTICLE
000197637 245__ $$aA survey and comparison of contemporary algorithms for computing the matrix geometric mean
000197637 269__ $$a2012
000197637 260__ $$c2012
000197637 336__ $$aJournal Articles
000197637 520__ $$aIn this paper we present a survey of various algorithms for computing matrix geometric means and derive new second-order optimization algorithms to compute the Karcher mean. These new algorithms are constructed using the standard definition of the Riemannian Hessian. The survey includes the ALM list of desired properties for a geometric mean, the analytical expression for the mean of two matrices, algorithms based on the centroid computation in Euclidean (flat) space, and Riemannian optimization techniques to compute the Karcher mean (preceded by a short introduction into differential geometry). A change of metric is considered in the optimization techniques to reduce the complexity of the structures used in these algorithms. Numerical experiments are presented to compare the existing and the newly developed algorithms. We conclude that currently first-order algorithms are best suited for this optimization problem as the size and/or number of the matrices increase. Copyright © 2012, Kent State University.
000197637 6531_ $$amatrix geometric mean
000197637 6531_ $$apositive definite matrices
000197637 6531_ $$aKarcher mean
000197637 6531_ $$aRiemannian optimization
000197637 700__ $$aJeuris, B.
000197637 700__ $$aVandebril, R.
000197637 700__ $$aVandereycken, B.
000197637 773__ $$j39$$q379-402$$tElectronic Transactions on Numerical Analysis
000197637 8564_ $$s408864$$uhttps://infoscience.epfl.ch/record/197637/files/pp379-402.pdf$$yn/a$$zn/a
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000197637 937__ $$aEPFL-ARTICLE-197637
000197637 973__ $$aEPFL$$rREVIEWED$$sPUBLISHED
000197637 980__ $$aARTICLE