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

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Abstract

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 Bačák, Jost, Sturm and others, and the lower curvature bound case seems entirely new.

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Acknowledgments

The authors would like to thank the anonymous referee for his valuable comments, in particular improving the discussion in Sect. 6. The second author would like to thank Prof. John Holbrook for raising his attention to the approximation problem of the barycenter treated in Remark 6.8 on the sphere. The second author had doubts in the convergence of such approximation scheme in the positive curvature case, but then he learned about the favorable outcomes of Prof. Holbrook’s numerical experiments on the sphere in a private communication with him, which initiated the further study of the problem.

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Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Shin-ichi Ohta & Miklós Pálfia

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  1. Shin-ichi Ohta
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  2. Miklós Pálfia
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Correspondence to Miklós Pálfia.

Additional information

Communicated by J. Jost.

S. Ohta is supported by the Grant-in-Aid for Young Scientists (B) 23740048; M. Pálfia is supported by the Research Fellowship of the Canon Foundation.

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Ohta, Si., Pálfia, M. Discrete-time gradient flows and law of large numbers in Alexandrov spaces. Calc. Var. 54, 1591–1610 (2015). https://doi.org/10.1007/s00526-015-0837-y

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