Discrete-time gradient flows and law of large numbers in Alexandrov spaces

详细信息    查看全文

• 作者：Shin-ichi Ohta ; Miklós Pálfia
• 关键词：51K05 ; 53C20 ; 58C05 ; 49L20 ; 49M37
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：54
• 期：2
• 页码：1591-1610
• 全文大小：559 KB
• 参考文献：1.Afsari, B.: Riemannian \(L^{p}\) center of mass: existence, uniqueness, and convexity. Proc. Am. Math. Soc. 139, 655-73 (2011)MathSciNet CrossRef MATH
  2.Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Birkh?user Verlag, Basel (2008)MATH
  3.Arnaudon, M., Li, X.M.: Barycenters of measures transported by stochastic flows. Ann. Probab. 33, 1509-543 (2005)MathSciNet CrossRef MATH
  4.Arnaudon, M., Dombry, C., Phan, A., Yang, L.: Stochastic algorithms for computing means of probability measures. Stoch. Process. Appl. 122, 1437-455 (2012)MathSciNet CrossRef MATH
  5.Ba?ák, M.: The proximal point algorithm in metric spaces. Israel J. Math. 194, 689-01 (2013)MathSciNet CrossRef MATH
  6.Ba?ák, M.: Computing means and medians in Hadamard spaces. SIAM J. Optim. (2014), to appear. arXiv:-210.-145
  7.Bento, G.C., Cruz, J.X.: Neto, finite termination of the proximal point method for convex functions on Hadamard manifolds. Optimization (2012). doi:10.-080/-2331934.-012.-30050
  8.Bertsekas, D.P.: Incremental proximal methods for large scale convex optimization. Math. Program. Ser. B 129, 163-95 (2011)MathSciNet CrossRef MATH
  9.Bertsekas, D.P., Tsitsiklis, J.N.: Neuro-dynamic Programming. Athena Scientific, Belmont (1996)MATH
  10.Bhatia, R.: Positive definite matrices. In: Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2007)
  11.Bhatia, R., Holbrook, J.: Riemannian geometry and matrix geometric means. Linear Algebra Appl. 413, 594-18 (2006)MathSciNet CrossRef MATH
  12.Brézis, H., Lions, P.-L.: Produits infinis de rèsolvantes. Israel J. Math. 29, 329-45 (1978)MathSciNet CrossRef MATH
  13.Burago, D., Burago, Yu., Ivanov, S.: A Course in Metric Geometry. American Mathematical Society, Providence (2001)CrossRef MATH
  14.Espínola, R., Fernández-León, A.: CAT\((k)\) -spaces, weak convergence and fixed points. J. Math. Anal. Appl. 353, 410-27 (2009)MathSciNet CrossRef MATH
  15.Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257-70 (2002)MathSciNet CrossRef MATH
  16.Holbrook, J.: No dice: a deterministic approach to the Cartan centroid. J. Ramanujan Math. Soc. 27, 509-21 (2012)MathSciNet MATH
  17.Jost, J.: Equilibrium maps between metric spaces. Calc. Var. Partial Differ. Equ. 2, 173-04 (1994)MathSciNet CrossRef MATH
  18.Jost, J.: Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70, 659-73 (1995)MathSciNet CrossRef MATH
  19.Jost, J.: Nonpositive Curvature: Geometric and Analytic Aspects. Birkh?user Verlag, Basel (1997)CrossRef MATH
  20.Jost, J.: Nonlinear dirichlet forms, new directions in Dirichlet forms 1-7. In: AMS/IP Stud. Adv. Math., vol. 8. American Mathematical Society, Providence (1998)
  21.Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509-41 (1977)MathSciNet CrossRef MATH
  22.Kendall, W.S.: Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence. Proc. Lond. Math. Soc. (3) 61, 371-06 (1990)MathSciNet CrossRef MATH
  23.Kendall, W.S.: Convexity and the hemisphere. J. Lond. Math. Soc. (2) 43, 567-76 (1991)MathSciNet CrossRef MATH
  24.Korf, L.A., Wets, R.J.-B.: Random lsc functions: an ergodic theorem. Math. Oper. Res. (26) 2, 421-45 (2001)MathSciNet CrossRef
  25.Kuwae, K.: Jensen’s inequality over CAT\((\kappa )\) -space with small diameter. In: Potential Theory and Stochastics in Albac. Theta Ser. Adv. Math., vol. 11, pp. 173-82. Theta, Bucharest (2009)
  26.Lawson, J., Lim, Y.: Monotonic properties of the least squares mean. Math. Ann. 351, 267-79 (2011)MathSciNet CrossRef MATH
  27.Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. (2) 79, 663-83 (2009)MathSciNet CrossRef MATH
  28.Lim, Y., Pálfia, M.: Weighted deterministic walks for the least squares mean on Hadamard spaces. Bull. Lond. Math. Soc. 46, 561-70 (2014)
  29.Lytchak, A.: Open map theorem for metric spaces. St. Petersb. Math. J. 17, 477-91 (2006)MathSciNet CrossRef MATH
  30.Mayer, U.F.: Gradient flows on nonpositively curved metric spaces and harmonic maps. Commun. Anal. Geom. 6, 199-53 (1998)MATH
  31.Moakher, M.: Means and averaging in the group of rotations. SIAM J. Matrix Anal. Appl. 24, 1-6 (2002)MathSciNet CrossRef MATH
  32.Nedic, A., Bertsekas, D.P.: Convergence rate of incremental subgradient algorithms. In: Stochastic Optimization: Algorithms and Applications (Gainesville, FL, 2000). Appl. Optim., vol. 54, pp. 223-64. Kluwer, Dordrecht (2001)
  33.Nedic, A., Bertsekas, D.P.: Incremental subgradient methods for nondifferentiable optimization. SIAM J. Optim. 12, 109-38 (2001)MathSciNet CrossRef MATH
  34.Ohta, S.: Convexities of metric spaces. Geom. Dedic. 1
• 作者单位：Shin-ichi Ohta (1)
  Miklós Pálfia (1)
  
  1. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

We develop the theory of discrete-time gradient flows for convex functions on Alexandrov spaces with arbitrary upper or lower curvature bounds. We employ different resolvent maps in the upper and lower curvature bound cases to construct such a flow, and show its convergence to a minimizer of the potential function. We also prove a stochastic version, a generalized law of large numbers for convex function valued random variables, which not only extends Sturm’s law of large numbers on nonpositively curved spaces to arbitrary lower or upper curvature bounds, but this version seems new even in the Euclidean setting. These results generalize those in nonpositively curved spaces (partly for squared distance functions) due to Ba?ák, Jost, Sturm and others, and the lower curvature bound case seems entirely new. Mathematics Subject Classification 51K05 53C20 58C05 49L20 49M37