Taiwanese Journal of Mathematics

NO DICE THEOREM ON SYMMETRIC CONES

Abstract

The monotonicity of the least squares mean on the Riemannian manifold of positive definite matrices, conjectured by Bhatia and Holbrook and one of key axiomatic properties of matrix geometric means, was recently established based on the Strong Law of Large Number [14, 4]. A natural question concerned with the S.L.L.N is so called the no dice conjecture. It is a problem to make a construction of deterministic sequences converging to the least squares mean without any probabilistic arguments. Very recently, Holbrook [7] gave an affirmative answer to the conjecture in the space of positive definite matrices. In this paper, inspired by the work of Holbrook [7] and the fact that the convex cone of positive definite matrices is a typical example of a symmetric cone (self-dual homogeneous convex cone), we establish the no dice theorem on general symmetric cones.

Article information

Source
Taiwanese J. Math., Volume 17, Number 6 (2013), 1967-1982.

Dates
First available in Project Euclid: 10 July 2017

https://projecteuclid.org/euclid.twjm/1499706280

Digital Object Identifier
doi:10.11650/tjm.17.2013.2944

Mathematical Reviews number (MathSciNet)
MR3141869

Zentralblatt MATH identifier
1295.47008

Citation

Kum, Sangho; Lee, Hosoo; Lim, Yongdo. NO DICE THEOREM ON SYMMETRIC CONES. Taiwanese J. Math. 17 (2013), no. 6, 1967--1982. doi:10.11650/tjm.17.2013.2944. https://projecteuclid.org/euclid.twjm/1499706280

References

• M. Baes, Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras, Linear Algebra Appl., 422 (2007), 664-700.
• H. H. Bauschke, O. Güler, A. S. Lewis and H. S. Sendov, Hyperbolic polynomials and convex analysis, Can. J. Math., 53 (2001), 470-488.
• R. Bhatia and J. Holbrook, Riemannian geometry and matrix geometric means, Linear Algebra Appl., 413 (2006), 594-618.
• R. Bhatia and R. Karandikar, Monotonicity of the matrix geometric mean, Math. Ann., 353 (2012), 1453-1467.
• J. Faraut and A. Korányi, Analysis on Symmetric Cones, Clarendon Press, Oxford, 1994.
• L. Faybusovich, Several Jordan-algebraic aspects of optimization, Optimization 57 (2008), 379-393.
• J. Holbrook, No dice: a determinstic approach to the Cartan centroid, J. Ramanujan Math. Soc. 27 (2012), 509-521.
• B. Jeuris, R. Vandebril and B. Vandereycken, A survey and comparison of contemporary algorithms for computing the matrix geometric mean, Electronic Transactions on Numerical Analysis, to appear.
• H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 30 (1977), 509-541.
• S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, Springer, 1999.
• J. D. Lawson and Y. Lim, The geometric mean, matrices, metrics, and more, Amer. Math. Monthly, 108 (2001), 797-812.
• J. D. Lawson and Y. Lim, A general framework for extending means to higher orders, Colloq. Math., 113 (2008), 191-221.
• J. Lawson, H. Lee and Y. Lim, Weighted geometric means, Forum Math., 24 (2012), 1067-1090.
• J. Lawson and Y. Lim, Monotonic properties of the least squares mean, Math. Ann., 351 (2011), 267-279.
• A. S. Lewis, The mathematics of eigenvalue optimization, Math. Program., Ser. B, 97 (2003), 155-176.
• Y. Lim, Geometric means on symmetric cones, Arch. der Math., 75 (2000), 39-45.
• Y. Lim, The Riemannian center of mass on symmetric cones of finite ranks, preprint.
• Y. Lim, J. Kim and L. Faybusovich, Simultaneous diagonalization on simple Euclidean Jordan algebras and its applications, Forum Math., 15 (2003), 639-644.
• M. Moakher, A differential geometric approach to the geometric mean of symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl., 26 (2005), 735-747.
• D. Sun and J. Sun, Löwner's operator and spectral functions on Euclidean Jordan algebras, Math. Oper. Res., 33 (2008), 421-445.
• K.-T. Sturm, Probability measures on metric spaces of nonpositive curvature, in: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, (P. Auscher et. al. Eds.), Contemp. Math. 338, Amer, Math. Soc. (AMS), Providence, 2003.
• J. Tao and M. S. Gowda, Some $P$-properties for nonlinear transformations on Euclidean Jordan algebras, Math. Oper. Res., 30 (2005), 985-1004.