Learning regularization parameters for general-form Tikhonov

Chung, Julianne; Español, Malena I.

doi:10.1088/1361-6420/33/7/074004

Cited by 37 publications

(60 citation statements)

References 40 publications

(98 reference statements)

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“…In inverse problems, the optimal inversion and experimental acquisition setup is discussed in the context of optimal model design in works by Haber, Horesh and Tenorio [25,26], as well as Ghattas et al [3,9]. Recently, parameter learning in the context of functional variational regularisation models (1.1) also entered the image processing community with works by the authors [10,22], Kunisch, Pock and co-workers [14,33], Chung et al [16] and Hintermüller et al [30].…”

Section: Introductionmentioning

confidence: 99%

Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

2016

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We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost based on a Huber-regularised TV seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a combined quasiNewton/semismooth Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between TGV 2 and ICTV is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.

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Section: Introductionmentioning

confidence: 99%

Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

2016

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“…However, (103) is sufficient to explain the majority of current state-of-the-art parameter learning approaches in the context of inverse problems. These cover the finitedimensional Markov random field models proposed in [325,346,143,124,334], the optimal model design approaches in [199,198,65,40], the optimal regularization parameter estimation in variational regularization [89,128,137,138,90,127], to training optimal operators in regularization functionals [123,122], reaction diffusion process [125,121], so-called variational networks [202,247,244] and other works related to image processing [301,214].…”

Section: Learningmentioning

confidence: 99%

Modern regularization methods for inverse problems

2018

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Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from linear towards nonlinear regularization methods even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this development towards modern nonlinear regularization methods, including their analysis, applications, and issues for future research.In particular we will discuss variational methods and techniques derived from those, since they have attracted particular interest in the last years and link to other fields like image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions, and learning theory.

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“…Definition 3. 5 We say that (y * , u * ) satisfies the second order sufficient optimality conditions of (P α,y δ ), if there exists λ * ∈ Z and η > 0 such that (y * , u * , λ * ) is a KKT point and…”

Section: Optimality Conditionsmentioning

confidence: 99%

A bilevel approach for parameter learning in inverse problems

2018

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A learning approach to selecting regularization parameters in multi-penalty Tikhonov regularization is investigated. It leads to a bilevel optimization problem, where the lower level problem is a Tikhonov regularized problem parameterized in the regularization parameters. Conditions which ensure the existence of solutions to the bilevel optimization problem of interest are derived, and these conditions are verified for two relevant examples. Difficulties arising from the possible lack of convexity of the lower level problems are discussed. Optimality conditions are given provided that a reasonable constraint qualification holds. Finally, results from numerical experiments used to test the developed theory are presented.Key words. parameter learning, Tikhonov regularization, bilevel optimization, multipenalty regularization where S is a mapping from a subset U ad of a Banach space U to a Banach space Y , the Tikhonov regularized problem consists in solving min u∈U ad J α,y δ (u) ≡ S(u) − y δ 2 + α · Ψ(u), (P α,y δ )

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Learning regularization parameters for general-form Tikhonov

Cited by 37 publications

References 40 publications

Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

Modern regularization methods for inverse problems

A bilevel approach for parameter learning in inverse problems

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