Convergence rates and source conditions for Tikhonov regularization with sparsity constraints

Details Convergence rates and source conditions for Tikhonov regularization with sparsity constraints Convergence rates and source conditions for Tikhonov regularization with sparsity constraints

2008-01-11

arxiv.org/abs/0801.1774v2

Journal of Inverse and Ill-Posed Problems, 16(5):463-478, 2008

Mathematics

Functional Analysis

Scientific paper

10.1515/JIIP.2008.025

This paper addresses the regularization by sparsity constraints by means of weighted $\ell^p$ penalties for $0\leq p\leq 2$. For $1\leq p\leq 2$ special attention is payed to convergence rates in norm and to source conditions. As main result it is proven that one gets a convergence rate in norm of $\sqrt{\delta}$ for $1\leq p\leq 2$ as soon as the unknown solution is sparse. The case $p=1$ needs a special technique where not only Bregman distances but also a so-called Bregman-Taylor distance has to be employed. For $p<1$ only preliminary results are shown. These results indicate that, different from $p\geq 1$, the regularizing properties depend on the interplay of the operator and the basis of sparsity. A counterexample for $p=0$ shows that regularization need not to happen.

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