Weak Sharp Minima on Riemannian Manifolds

Modern signal and image acquisition systems are able to capture data that is no longer real-valued, but may take values on a manifold. However, whenever measurements are taken, no matter whether manifold-valued or not, there occur tiny inaccuracies, which result in noisy data. In this chapter, we review recent advances in denoising of manifoldvalued signals and images, where we restrict our attention to variational models and appropriate minimization algorithms. The algorithms are either classical as the subgradient algorithm or generalizations of the half-quadratic minimization method, the cyclic proximal point algorithm, and the Douglas-Rachford algorithm to manifolds. An important aspect when dealing with real-world data is the practical implementation. Here several groups provide software and toolboxes as the Manifold Optimization (Manopt) package and the manifold-valued image restoration toolbox (MVIRT).

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“…see, e.g., [45] or [59] for finite functions ϕ. For any x ∈ int(dom ϕ), the subdifferential is a nonempty convex and compact set in T…”

Section: Subgradient Descentmentioning

confidence: 99%

Recent advances in denoising of manifold-valued images

Bergmann

Laus

Persch

et al. 2019

Handbook of Numerical Analysis

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“…Furthermore, by [34,Example 6.2], S is the set of local weak sharp minima for problem (5.2). Hence, Corollary 5.9 is applicable to concluding that any sequence {x n }, generated by Algorithm IP M with {δ n } satisfying δ n < +∞ and the parameters {λ n } satisfying (4.15), terminates in a finite number of iterations.…”

Section: Numerical Examplesmentioning

confidence: 99%

Proximal Point Algorithms on Hadamard Manifolds: Linear Convergence and Finite Termination

Wang¹,

Li²,

López-Acedo³

et al. 2016

SIAM J. Optim.

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Abstract. In the present paper, we consider inexact proximal point algorithms for finding singular points of multivalued vector fields on Hadamard manifolds. The rate of convergence is shown to be linear under the mild assumption of metric subregularity. Furthermore, if the sequence of parameters associated with the iterative scheme converges to 0, then the convergence rate is superlinear. At the same time, the finite termination of the inexact proximal point algorithm is also provided under a weak sharp minima-like condition. Applications to optimization problems are provided. Some of our results are new even in Euclidean spaces, while others improve and/or extend some known results in Euclidean spaces. As a matter of fact, in the case of exact proximal point algorithm, our results improve the corresponding results in [G. C. Bento and J. X. Cruz Neto, Optim., 63 (2014), pp. 1281-1288. Finally, several examples are provided to illustrate that our results are applicable while the corresponding results in the Hilbert space setting are not.

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“…Note that in a number of papers [3,6,21,28,37,40] contain either necessary or sufficient conditions of the dual-type for weak sharp minima given in terms of some normal cones and subdifferentials. In particular, the necessity part of (ii), and hence of (iii) and (iv), is proved in [27] for the general Banach space setting.…”

Section: !T*(z-u\ii~-=-~d ~ (T*(z-u\of(u))mentioning

confidence: 99%

“…Furthermore, close relationships between weak sharp minima, linear regularity, metric regularity, and error bound were exploited in [4,5]. The recent paper [21] considers weak sharp minima for convex constrained optimization problems on Riemannian manifolds, containing also new characterizations for the case of conventional convex problems in finite-dimensional spaces.…”

Section: Introductionmentioning

confidence: 99%

Complete characterizations of local weak sharp minima with applications to semi-infinite optimization and complementarity

Zhou

Mordukhovich

Xiu

2012

Nonlinear Analysis: Theory, Methods & Applications

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In this paper we identify a favorable class of nonsmooth functions for which local weak sharp minima can be completely characterized in terms of normal cones and subdifferentials, or tangent cones and subderivatives, or their mixture in finite-dimensional spaces. The results obtained not only significantly extend previous ones in the literature, but also allow us to provide new types of criteria for local weak sharpness. Applications of the developed theory are given to semi-infinite programming and to semi-infinite complementarity problems.

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Weak Sharp Minima on Riemannian Manifolds

Cited by 94 publications

References 58 publications

Recent advances in denoising of manifold-valued images

Recent advances in denoising of manifold-valued images

Proximal Point Algorithms on Hadamard Manifolds: Linear Convergence and Finite Termination

Complete characterizations of local weak sharp minima with applications to semi-infinite optimization and complementarity

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