A survey and comparison of contemporary algorithms for computing the matrix geometric mean
A survey and comparison of contemporary algorithms for computing the matrix geometric mean
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DOI：
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发表时间：
2012
影响因子：
1.3
通讯作者：
Ben Jeuris;R. Vandebril;Bart Vandereycken
中科院分区：
文献类型：
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作者：
Ben Jeuris;R. Vandebril;Bart Vandereycken
在本文中，我们提出了一个调查的各种算法计算矩阵的几何平均值，并推导出新的二阶优化算法计算的Karcher平均值。这些新的算法构造使用的标准定义的黎曼海森。该调查包括ALM列表的期望属性的几何平均值，分析表达式的平均值的两个矩阵，算法的基础上的质心计算在欧几里德（平坦）空间，和黎曼优化技术计算的Karcher平均值（前面有一个简短的介绍微分几何）。在优化技术中考虑了度量的变化，以降低这些算法中使用的结构的复杂性。数值实验比较现有的和新开发的算法。我们的结论是，目前一阶算法是最适合这个优化问题的矩阵的大小和/或数量的增加。版权所有© 2012，肯特州立大学。
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.