Tikhonov regularization with \({\ell^{0}}\)-term complementing a convex penalty: \({\ell^{1}}\)-convergence under sparsity constraints

DOI10.1515/jiip-2019-0008zbMath1480.65139arXiv1810.08775OpenAlexW2963383658MaRDI QIDQ2316708

Wei Wang, Shuai Lu, Jin Cheng, Bernd Hofmann

Publication date: 6 August 2019

Published in: Journal of Inverse and Ill-Posed Problems (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1810.08775

zbMATH Keywords

convergence ratesTikhonov regularizationlinear ill-posed problemssparsity constraintsapproximate source conditions\({\ell^{0}}\)-penalty term\({\ell^{1}}\)-error estimates

Mathematics Subject Classification ID

Numerical solutions of ill-posed problems in abstract spaces; regularization (65J20) Linear operators and ill-posed problems, regularization (47A52)

Related Items

Proximal algorithm for minimization problems in \(l_0\)-regularization for nonlinear inverse problems, Sparse regularization with the ℓ0 norm, Morozov's discrepancy principle for \(\alpha\ell_1-\beta\ell_2\) sparsity regularization, $ \newcommand{\e}{{\rm e}} {\alpha\ell_{1}-\beta\ell_{2}}$ regularization for sparse recovery, Variational Source Conditions in $L^p$-spaces

Cites Work

• Unnamed Item
• Unnamed Item
• Convex Tikhonov regularization in Banach spaces: new results on convergence rates
• Regularization methods in Banach spaces.
• An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems
• Convergence rates for regularization with sparsity constraints
• Regularization properties of Tikhonov regularization with sparsity constraints
• Non-convex sparse regularisation
• Error estimation for Bregman iterations and inverse scale space methods in image restoration
• Variational methods in imaging
• Direct and inverse results in variable Hilbert scales
• Statistical and computational inverse problems.
• Well-posedness and convergence rates for sparse regularization with sublinear \(l^q\) penalty term
• The index function and Tikhonov regularization for ill-posed problems
• Variational smoothness assumptions in convergence rate theory—an overview
• Multi-parameter Tikhonov regularization with the ℓ ⁰ sparsity constraint
• Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces
• Elastic-net regularization versus ℓ ¹ -regularization for linear inverse problems with quasi-sparse solutions
• On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems
• Generalized Bregman distances and convergence rates for non-convex regularization methods
• Sparse and stable Markowitz portfolios
• Parameter choice in Banach space regularization under variational inequalities
• Regularization with non-convex separable constraints
• Convergence rates and source conditions for Tikhonov regularization with sparsity constraints
• Modulus of continuity for conditionally stable ill-posed problems in Hilbert space
• Sparse regularization with l ^q penalty term
• Approximate source conditions for nonlinear ill-posed problems—chances and limitations
• Handbook of Mathematical Methods in Imaging
• Elastic-net regularization: error estimates and active set methods
• One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization
• Convergence rates of convex variational regularization
• Convergence rates inℓ¹-regularization if the sparsity assumption fails
• Convergence rates inℓ¹-regularization when the basis is not smooth enough
• A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators
• Regularization and Variable Selection Via the Elastic Net
• A variational approach to sparsity optimization based on Lagrange multiplier theory
• On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization
• Approximate source conditions in Tikhonov–Phillips regularization and consequences for inverse problems with multiplication operators
• Regularization of ill-posed problems in Banach spaces: convergence rates