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

Ohta, Shin‐ichi; Pálfia, Miklós

doi:10.1007/s00526-015-0837-y

Cited by 22 publications

(23 citation statements)

References 41 publications

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“…which is called the proximal point algorithm [1,9,18], where {λ n } is a sequence of positive real numbers. The convergence of the proximal point algorithm in a metric space (e.g.…”

Section: Introductionmentioning

confidence: 99%

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The Proximal Point Algorithm in Uniformly Convex Metric Spaces

Choi¹,

Ji²

2016

Communications of the Korean Mathematical Society

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“…which is called the proximal point algorithm [1,9,18], where {λ n } is a sequence of positive real numbers. The convergence of the proximal point algorithm in a metric space (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…The convergence of the proximal point algorithm in a metric space (e.g. CAT(0)-spaces and Alexandrov spaces) have been studied by several authors [1,9,18], etc.…”

Section: Introductionmentioning

confidence: 99%

The Proximal Point Algorithm in Uniformly Convex Metric Spaces

Choi¹,

Ji²

2016

Communications of the Korean Mathematical Society

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“…Later a deterministic, also called "nodice", version of Sturm's law that periodically recycles all the points A i was proved by Holbrook [12] in P and then in the CAT(0) metric setting by the authors [22], and independently in [4]. This "nodice" theorem states that S n converges to Λ( k−1 i=0 1 k δ Ai ) for the recycling deterministic version Y n := A n in (2), where n denotes the residual of n modulo k. Further generalizations were proved in the CAT (1) setting by Ohta and the second author [28] and by Yokota [36]. These metric approaches treat the sequence of inductive means S n as a discrete-time approximation of the gradient flow of the cost function to be minimized in (1), and apply the Riemannian-like nature of the CAT (1) property in an essential way.…”

Section: The Contractive Barycenter Of Positive Operatorsmentioning

confidence: 98%

Strong law of large numbers for the $L^1$-Karcher mean

Lim¹,

Pálfia²

2019

Preprint

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Sturm's strong law of large numbers in CAT(0) spaces has been an influential tool to study the geometric mean or also called Karcher barycenter of positive definite matrices. It provides an easily computable stochastic approximation based on inductive means. Convergence of a deterministic version of this approximation has been proved by Holbrook, providing his "nodice" theorem for the Karcher mean of positive definite matrices. The Karcher mean has also been extended to the infinite dimensional case of positive operators on a Hilbert space by Lawson-Lim and then to probability measures with bounded support by the second author, however the CAT(0) property of the space is lost and one defines the mean as the unique solution of a nonlinear operator equation on a convex Banach-Finsler manifold. The formulations of Sturm's strong law of large numbers and Holbrook's "nodice" approximation are natural and both conjectured to converge, however all previous techniques of their proofs break down, due to the Banach-Finsler nature of the space. In this paper we prove both conjectures by establishing the most general L 1 -form of Sturm's strong law of large numbers and Holbrook's "nodice" theorem in the operator norm by developing a stochastic discrete-time resolvent flow for the Karcher barycenter using its Wasserstein contraction property.

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“…Moreover Perel'man and Petrunin [42], [43] refined the theory into semiconcave functions on Alexandrov spaces. Their studies have been succeeded to many other researches such as [13], [14], [34], [35], [36], [38], etc.…”

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confidence: 95%

Approximations of Lipschitz maps via Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions

Kondo¹

2018

Preprint

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We show, as our main theorem, that if a Lipschitz map from a compact Riemannian manifold M to a connected compact Riemannian manifold N , where dim M ≥ dim N , has no singular points on M in the sense of F.H. Clarke, then the map admits a smooth approximation via Ehresmann's fibrations. We also show the Reeb sphere theorem for a closed Riemannian manifold M of m := dim M ≥ 2 which admits a Lipschitz function F : M −→ R with only two singular points, denoted by z 1 , z 2 ∈ M , in the sense of Clarke by assuming that there is a constant c between F (z 1 ) and F (z 2 ) such that F −1 (c) is homeomorphic to an (m − 1)dimensional sphere. In the proof of our sphere theorem we use a corollary arising from the process of the proof of the main theorem above.

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

Cited by 22 publications

References 41 publications

The Proximal Point Algorithm in Uniformly Convex Metric Spaces

The Proximal Point Algorithm in Uniformly Convex Metric Spaces

Strong law of large numbers for the $L^1$-Karcher mean

Approximations of Lipschitz maps via Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions

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