Incremental gradient method for Karcher mean on symmetric cones. (English) Zbl 1432.65056
Summary: In this paper, we deal with the minimization problem for computing Karcher mean on a symmetric cone. The objective of this minimization problem consists of the sum of squares of the Riemannian distances with many given points in a symmetric cone. Moreover, the problem can be reduced to a bound-constrained minimization problem. These motivate us to adapt an incremental gradient method. So we propose an incremental gradient method and establish its global convergence properties exploiting the Lipschitz continuity of the gradient of the Riemannian distance function.
MSC:
| 65F99 | Numerical linear algebra |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 65K05 | Numerical mathematical programming methods |
| 90C25 | Convex programming |
Software:
Matrix Means ToolboxReferences:
| [1] | Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509-541 (1977) · Zbl 0354.57005 · doi:10.1002/cpa.3160300502 |
| [2] | Afsari, B., Tron, R., Vidal, R.: On the convergence of gradient descent for finding the Riemannian center of mass. SIAM J. Control Optim. 51, 2230-2260 (2013) · Zbl 1285.90031 · doi:10.1137/12086282X |
| [3] | Bacak, M.: Computing medians and means in Hadamard spaces. SIAM J. Optim. 24, 1542-1566 (2014) · Zbl 1306.49046 · doi:10.1137/140953393 |
| [4] | Bini, D.A., Iannazzo, B.: Computing the Karcher mean of symmetric positive definite matrices. Linear Algebra Appl. 438, 1700-1710 (2013) · Zbl 1268.15007 · doi:10.1016/j.laa.2011.08.052 |
| [5] | Gregorio, R. M., Oliveira, P. R.: A proximal technique for computing the Karcher mean of symmetric positive definite matrices. Preprint (2013) · Zbl 1325.47039 |
| [6] | Jeuris, B., Vandebril, R., Vandereycken, B.: A survey and comparison of contemporary algorithms for computing the matrix geometric mean. Electron. Trans. Numer. Anal. 39, 379-402 (2012) · Zbl 1287.65036 |
| [7] | Tseng, P., Yun, S.: Incrementally updated gradient methods for constrained and regularized optimization. J. Optim. Theory Appl. 160, 832-853 (2014) · Zbl 1300.90050 · doi:10.1007/s10957-013-0409-2 |
| [8] | Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Clarendon Press, Oxford (1994) · Zbl 0841.43002 |
| [9] | Sun, D., Sun, J.: Löwner’s operator and spectral functions on Euclidean Jordan algebras. Math. Oper. Res. 33, 421-445 (2008) · Zbl 1218.90197 · doi:10.1287/moor.1070.0300 |
| [10] | Kum, S., Lim, Y.: Penalized complementarity functions on symmetric cones. J. Glob. Optim. 46, 475-485 (2010) · Zbl 1206.90192 · doi:10.1007/s10898-009-9450-y |
| [11] | Kum, S., Lee, H., Lim, Y.: No dice theorem on symmetric cones. Taiwan. J. Math. 17, 1967-1982 (2013) · Zbl 1295.47008 |
| [12] | Lang, S.: Fundamentals of Differential Geometry. Graduate Texts in Mathematics. Springer, New York (1999) · Zbl 0932.53001 · doi:10.1007/978-1-4612-0541-8 |
| [13] | Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, 663-683 (2009) · Zbl 1171.58001 · doi:10.1112/jlms/jdn087 |
| [14] | Bertsekas, D. P.: Incremental gradient, subgradient, and proximal methods for convex optimization: A Survey. Lab. for Information and Decision Systems Report LIDS-P-2848, MIT; arXiv:1507.01030 (2010) |
| [15] | Rothaus, O.S.: Domains of positivity. Abh. Math. Semin. Univ. Hambg. 24, 189-235 (1960) · Zbl 0096.27903 · doi:10.1007/BF02942030 |
| [16] | Holbrook, J.: No dice: a deterministic approach to the Cartan centroid. J. Ramanujan Math. Soc. 27, 509-521 (2012) · Zbl 1325.47039 |
| [17] | Baes, M.: Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras. Linear Algebra Appl. 422, 664-700 (2007) · Zbl 1138.90018 · doi:10.1016/j.laa.2006.11.025 |
| [18] | Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999) · Zbl 1015.90077 |
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