Discrete-time gradient flows and law of large numbers in Alexandrov spaces
Discrete-time gradient flows and law of large numbers in Alexandrov spaces
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DOI：
10.1007/s00526-015-0837-y
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发表时间：
2014-02
影响因子：
2.1
通讯作者：
Shin-ichi Ohta;Miklós Pálfia
中科院分区：
文献类型：
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作者：
Shin-ichi Ohta;Miklós Pálfia
我们发展了Alexandrov空间上的凸函数的离散时间梯度流理论，其曲率上界或下界是任意的。我们采用不同的预解映射的上下曲率界的情况下，构建这样的流，并显示其收敛到一个最小的潜在功能。我们还证明了一个随机版本，凸函数值随机变量的广义大数定律，它不仅扩展了Sturm的大数定律的非正弯曲的空间，以任意的曲率下限或上限，但这个版本似乎新的，即使在欧几里德设置。这些结果推广了Bačák、Jost、Sturm等人在非正曲空间（部分是平方距离函数）中的结果，而下曲率界的情况似乎是全新的。
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.