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Licensed Unlicensed Requires Authentication Published by De Gruyter September 12, 2008

Convergence rates and source conditions for Tikhonov regularization with sparsity constraints

  • D. A. Lorenz

Abstract

This paper addresses the regularization by sparsity constraints by means of weighted ℓp penalties for 0 ≤ p ≤ 2. For 1 ≤ p ≤ 2 special attention is payed to convergence rates in norm and to source conditions. As main results it is proven that one gets a convergence rate of in the 2-norm for 1 < p ≤ 2 and in the 1-norm for p = 1 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 ≥ 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.

Received: 2008-01-25
Revised: 2008-04-23
Published Online: 2008-09-12
Published in Print: 2008-August

© de Gruyter 2008

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