The Riemann Barycenter Computation and Means of Several Matrices
An iterative definition of any n variable mean function is given in this article, which iteratively uses the two-variable form of the corresponding two-variable mean function. This extension method omits recursivity which is an important improvement compared with certain recursive formulas given before by Ando-Li-Mathias, Petz- Temesi. Furthermore it is conjectured here that this iterative algorithm coincides with the solution of the Riemann centroid minimization problem. Certain simulations are given here to compare the convergence rate of the different algorithms given in the literature. These algorithms will be the gradient and the Newton mehod for the Riemann centroid computation.
[1] E. Ahn, S. Kim and Y. Lim, An extended Lie-Trotter formula and its
applications, Linear Algebra and its Appl., 427 (2007), pp. 190-196.
[2] T. Ando, C-K. Li and R. Mathias, Geometric means, Linear Alg. Appl.,
385 (2004), pp. 305-334.
[3] W. N. Anderson Jr., Shorted operators, SIAM J. Appl. Math., 20 (1971),
pp. 520-525.
[4] W. N. Jr. Anderson and R. J. Duffin, Series and parallel addition of
matrices, J. Math. Anal. Appl., 26 (1969), pp. 576-594.
[5] W. N. Jr. Anderson and G. E. Trapp, Shorted Operators II, SIAM J. Appl.
Math., 28 (1975), pp. 60-71.
[6] J. Antezanaa, G. Corachb and D. Stojanoff, Bilateral shorted operators
and parallel sums, Linear Alg. Appl., 414 (2006), pp. 570-588.
[7] V. Arsigny, P. Fillard, X. Pennec and N. Ayache, Geometric Means in
a Novel Vector Space Structure on Symmetric Positive-Definite Matrices,
SIAM J. Matrix Anal. Appl., 29 (2007), pp. 328-347.
[8] R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1996.
[9] R. Bhatia and J. Holbrook, Riemannian geometry and matrix geometric
means, Linear Alg. Appl., 413 (2006), pp. 594618.
[10] R. Ferreira and J. Xavier, Hessian of the Riemannian squared-distance
function on connected locally symmetric spaces with applications, Controlo
2006, 7th Portuguese Conference on Automatic Control , special
session on control, optimization and computation, September 2006.
[11] P. Fillard, V. Arsigny, X. Pennec and N. Ayache, Joint estimation and
smoothing of clinical DT-MRI with a Log-Euclidean metric. Research
Report RR-5607, INRIA, Sophia-Antipolis, France, June 2005.
[12] F. Hiai and H. Kosaki, Means of Hilbert space operators, Lecture Notes
In Maths. 1820 (2003), Springer, .
[13] H. Karcher, Riemannian center of mass and mollifier smoothing, Comm.
Pure Appl. Math., vol. 30 (1977), pp. 509541.
[14] F. Kubo and T. Ando, Means of positive linear operators, Math. Ann.,
246 (1980), pp. 205-224.
[15] J. H. Manton, A globally convergent numerical algorithm for computing
the centre of mass on compact Lie groups, in Eighth International
Conference on Control, Automation, Robotics and Vision, Kunming,
China, December 2004.
[16] M. Moakher, A Differential Geometric Approach to the Geometric Mean
of Symmetric Positive-Definite Matrices, SIAM J. Matrix Anal. Appl., 26
(2005), pp. 735-747.
[17] M. Moakher, Means and averaging in the group of rotations, SIAM J.
Matrix Anal. Appl., vol. 24, no. 1 (2002), pp. 116.
[18] M. P'alfia, Algorithmic definition of means acting on positive numbers
and operators, submitted (arXiv:math/0503510).
[19] M. P'alfia, Means of unorderable matrices and the Riemann centroid,
submitted to Linear Algebra and its Applications.
[20] D. Petz and R. Temesi, Means of positive numbers and matrices, SIAM
J. Matrix Anal. Appl., 27 (2006), pp. 712-720.
[1] E. Ahn, S. Kim and Y. Lim, An extended Lie-Trotter formula and its
applications, Linear Algebra and its Appl., 427 (2007), pp. 190-196.
[2] T. Ando, C-K. Li and R. Mathias, Geometric means, Linear Alg. Appl.,
385 (2004), pp. 305-334.
[3] W. N. Anderson Jr., Shorted operators, SIAM J. Appl. Math., 20 (1971),
pp. 520-525.
[4] W. N. Jr. Anderson and R. J. Duffin, Series and parallel addition of
matrices, J. Math. Anal. Appl., 26 (1969), pp. 576-594.
[5] W. N. Jr. Anderson and G. E. Trapp, Shorted Operators II, SIAM J. Appl.
Math., 28 (1975), pp. 60-71.
[6] J. Antezanaa, G. Corachb and D. Stojanoff, Bilateral shorted operators
and parallel sums, Linear Alg. Appl., 414 (2006), pp. 570-588.
[7] V. Arsigny, P. Fillard, X. Pennec and N. Ayache, Geometric Means in
a Novel Vector Space Structure on Symmetric Positive-Definite Matrices,
SIAM J. Matrix Anal. Appl., 29 (2007), pp. 328-347.
[8] R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1996.
[9] R. Bhatia and J. Holbrook, Riemannian geometry and matrix geometric
means, Linear Alg. Appl., 413 (2006), pp. 594618.
[10] R. Ferreira and J. Xavier, Hessian of the Riemannian squared-distance
function on connected locally symmetric spaces with applications, Controlo
2006, 7th Portuguese Conference on Automatic Control , special
session on control, optimization and computation, September 2006.
[11] P. Fillard, V. Arsigny, X. Pennec and N. Ayache, Joint estimation and
smoothing of clinical DT-MRI with a Log-Euclidean metric. Research
Report RR-5607, INRIA, Sophia-Antipolis, France, June 2005.
[12] F. Hiai and H. Kosaki, Means of Hilbert space operators, Lecture Notes
In Maths. 1820 (2003), Springer, .
[13] H. Karcher, Riemannian center of mass and mollifier smoothing, Comm.
Pure Appl. Math., vol. 30 (1977), pp. 509541.
[14] F. Kubo and T. Ando, Means of positive linear operators, Math. Ann.,
246 (1980), pp. 205-224.
[15] J. H. Manton, A globally convergent numerical algorithm for computing
the centre of mass on compact Lie groups, in Eighth International
Conference on Control, Automation, Robotics and Vision, Kunming,
China, December 2004.
[16] M. Moakher, A Differential Geometric Approach to the Geometric Mean
of Symmetric Positive-Definite Matrices, SIAM J. Matrix Anal. Appl., 26
(2005), pp. 735-747.
[17] M. Moakher, Means and averaging in the group of rotations, SIAM J.
Matrix Anal. Appl., vol. 24, no. 1 (2002), pp. 116.
[18] M. P'alfia, Algorithmic definition of means acting on positive numbers
and operators, submitted (arXiv:math/0503510).
[19] M. P'alfia, Means of unorderable matrices and the Riemann centroid,
submitted to Linear Algebra and its Applications.
[20] D. Petz and R. Temesi, Means of positive numbers and matrices, SIAM
J. Matrix Anal. Appl., 27 (2006), pp. 712-720.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:49906", author = "Miklos Palfia", title = "The Riemann Barycenter Computation and Means of Several Matrices", abstract = "An iterative definition of any n variable mean function is given in this article, which iteratively uses the two-variable form of the corresponding two-variable mean function. This extension method omits recursivity which is an important improvement compared with certain recursive formulas given before by Ando-Li-Mathias, Petz- Temesi. Furthermore it is conjectured here that this iterative algorithm coincides with the solution of the Riemann centroid minimization problem. Certain simulations are given here to compare the convergence rate of the different algorithms given in the literature. These algorithms will be the gradient and the Newton mehod for the Riemann centroid computation.
", keywords = "Means, matrix means, operator means, geometric mean, Riemannian center of mass.", volume = "2", number = "4", pages = "230-6", }
