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arXiv · 0806.3222

Sparse Regularization with $l^q$ Penalty Term

Markus Grasmair·Markus Haltmeier·Otmar Scherzer
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Abstract

We consider the stable approximation of sparse solutions to non-linear 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 $O(\sqrtδ)$ 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 $O(δ)$.

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Markus Grasmair, Markus Haltmeier, Otmar Scherzer. 2008-08-25. Sparse Regularization with $l^q$ Penalty Term. https://doi.org/10.1088/0266-5611%2F24%2F5%2F055020

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