Convergence rates in $\ell^1$-regularization when the basis is not smooth enough

Jens Flemming; Markus Hegland

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arXiv · 1311.1923

Convergence rates in $\ell^1$-regularization when the basis is not smooth enough

Jens Flemming·Markus Hegland

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Abstract

Sparsity promoting regularization is an important technique for signal reconstruction and several other ill-posed problems. Theoretical investigation typically bases on the assumption that the unknown solution has a sparse representation with respect to a fixed basis. We drop this sparsity assumption and provide error estimates for non-sparse solutions. After discussing a result in this direction published earlier by one of the authors and coauthors we prove a similar error estimate under weaker assumptions. Two examples illustrate that this set of weaker assumptions indeed covers additional situations which appear in applications.

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Jens Flemming, Markus Hegland. 2013-11-08. Convergence rates in $\ell^1$-regularization when the basis is not smooth enough. https://arxiv.org/abs/1311.1923

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