Sparse regularization with lq penalty term
By Markus Grasmair, Markus Haltmeier, Otmar Scherzer. 2008
137 citations recorded by the source corpus.
We consider the stable approximation of sparse solutions to nonlinear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate of the regularized solutions in dependence of the noise level δ. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to O(δ).
Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/74edfe113c56f745b182954e805e0d76d1c677b4
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Computational Mechanics
- Peter Maass
- Bangti Jin
- Markus Haltmeier
- Markus Grasmair
- Otmar Scherzer
- Daniel Wachsmuth
- Gerd Wachsmuth
- Stephan Antholzer
In Computer Vision and Pattern Recognition
In Mathematical Physics
- Peter Maass
- Bangti Jin
- Markus Haltmeier
- Markus Grasmair
- Otmar Scherzer
- Daniel Wachsmuth
- Gerd Wachsmuth
- Stephan Antholzer
Citations in the corpus
- NETT: solving inverse problems with deep neural networks (2018, background): “λ∈Λ and NETT regularization reduces to sparse l-regularization [17, 22, 23, 34, 41].”
- An analysis of electrical impedance tomography with applications to Tikhonov regularization (2012, methodology): “This can be remedied by imposing extra conditions [24, 46].”
- Convergence and regularization results for optimal control problems with sparsity functional (2011, background): “Other source conditions as developed in [10,21] are not directly applicable, since they are tailored to the case (α, β) → (0, 0), while we are considering the convergence for α → 0 and fixed β.”
- Necessary and sufficient conditions for linear convergence of ℓ1‐regularization (2011, methodology): “In addition, we compare these results with results on general convergence rates with respect to the Bregman distance (see [6, 23, 24, 25, 26, 28, 29, 31, 32, 34]).”