Convergence rates for ℓ1-regularization without injectivity-type assumptions. (5th July 2016) - Flemming, Jens
Abstract: Existing convergence rate results for sparsity promoting regularization of Tikhonov-type rely on injectivity of the considered operator or at least on slightly weakened injectivity assumptions (finite basis injectivity or restricted isometry property). We extend such results to non-injective operators by formulating a suitable variational source condition, which then is characterized in the language of range conditions with respect to the range of the adjoint operator. As a special case we consider operator equations with uniquely determined 1-norm minimizing solutions. Based on the developed characterization of a variational source condition we also provide convergence rates for the case that solutions are not sparse.