Convex analysis and optimization in Hadamard spaces

Bacak, Miroslav

doi:10.1515/9783110361629

Cited by 151 publications

(87 citation statements)

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“…Turning to more applied topics, [Bač14b] shows algorithms for calculating the Fréchet mean in Hadamard spaces with cost function d 2a for a = 1 2 and a = 1. An important application of Hadamard spaces in Bioinformatics are phylogenetic trees [BHV01].…”

Section: Hadamard Spacesmentioning

confidence: 99%

Convergence rates for the generalized Fréchet mean via the quadruple inequality

Schötz¹

2019

Electron. J. Statist.

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For sets Q and Y, the generalized Fréchet mean m ∈ Q of a random variable Y , which has values in Y, is any minimizerThere are little restrictions to Q and Y. In particular, Q can be a non-Euclidean metric space. We provide convergence rates for the empirical generalized Fréchet mean. Conditions for rates in probability and rates in expectation are given. In contrast to previous results on Fréchet means, we do not require a finite diameter of the Q or Y. Instead, we assume an inequality, which we call quadruple inequality. It generalizes an otherwise common Lipschitz condition on the cost function. This quadruple inequality is known to hold in Hadamard spaces. We show that it also holds in a suitable way for certain powers of a Hadamard-metric.

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Section: Hadamard Spacesmentioning

confidence: 99%

Convergence rates for the generalized Fréchet mean via the quadruple inequality

Schötz¹

2019

Electron. J. Statist.

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“…However, when restricting to the case of Hadamard manifolds, i.e., manifolds of nonpositive sectional curvature, all important properties carry over, i.e., convexity and even lower semi-continuity and coerciveness. For details on optimization on Hadamard manifolds we refer to the monograph [10]. In the case of Hadamard manifolds we may thus conclude the existence of respective minimizers f ∈ H(V ; M) of (4) and (6).…”

Section: Optimality Conditionsmentioning

confidence: 89%

A Graph Framework for Manifold-Valued Data

Bergmann¹,

Tenbrinck²

2018

SIAM J. Imaging Sci.

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Graph-based methods have been proposed as a unified framework for discrete calculus of local and nonlocal image processing methods in the recent years. In order to translate variational models and partial differential equations to a graph, certain operators have been investigated and successfully applied to real-world applications involving graph models. So far the graph framework has been limited to real-and vector-valued functions on Euclidean domains. In this paper we generalize this model to the case of manifold-valued data. We introduce the basic calculus needed to formulate variational models and partial differential equations for manifold-valued functions and discuss the proposed graph framework for two particular families of operators, namely, the isotropic and anisotropic graph p-Laplacian operators, p ≥ 1. Based on the choice of p we are in particular able to solve optimization problems on manifold-valued functions involving total variation (p = 1) and Tikhonov (p = 2) regularization. Finally, we present numerical results from processing both synthetic as well as real-world manifold-valued data, e.g., from diffusion tensor imaging (DTI) and light detection and ranging (LiDAR) data. * Felix-Klein-Zentrum,

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“…In particular, Goebel and Reich [11] introduced the hyperbolic metric on the open unit ball of a complex Hilbert space, from a viewpoint of holomorphic function theory. They developed the study of the Hilbert ball, which leads to research onconvexity, nonexpansive mappings, fixed point theorems, and so forth, and [11] is cited in many bibliography such as [12]. The definition of is equivalent to…”

Section: The Poincaré Metric and Orthogonal Gyroexpansion In The Möbimentioning

confidence: 99%

“…The author would like to thank Professor Michio Seto for bringing information on literatures [11,12].…”

Section: Acknowledgmentsmentioning

confidence: 99%

Orthogonal Gyroexpansion in Möbius Gyrovector Spaces

Watanabe

2017

Journal of Function Spaces

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We investigate the Möbius gyrovector spaces which are open balls centered at the origin in a real Hilbert space with the Möbius addition, the Möbius scalar multiplication, and the Poincaré metric introduced by Ungar. In particular, for an arbitrary point, we can easily obtain the unique closest point in any closed gyrovector subspace, by using the ordinary orthogonal decomposition. Further, we show that each element has the orthogonal gyroexpansion with respect to any orthogonal basis in a Möbius gyrovector space, which is similar to each element in a Hilbert space having the orthogonal expansion with respect to any orthonormal basis. Moreover, we present a concrete procedure to calculate the gyrocoefficients of the orthogonal gyroexpansion.

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Convex analysis and optimization in Hadamard spaces

Cited by 151 publications

References 0 publications

Convergence rates for the generalized Fréchet mean via the quadruple inequality

Convergence rates for the generalized Fréchet mean via the quadruple inequality

A Graph Framework for Manifold-Valued Data

Orthogonal Gyroexpansion in Möbius Gyrovector Spaces

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