Sparse regularization of inverse problems by operator-adapted frame thresholding

Jürgen Frikel; Markus Haltmeier

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arXiv · 1909.09364

Sparse regularization of inverse problems by operator-adapted frame thresholding

Jürgen Frikel·Markus Haltmeier

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Abstract

We analyze sparse frame based regularization of inverse problems by means of a diagonal frame decomposition (DFD) for the forward operator, which generalizes the SVD. The DFD allows to define a non-iterative (direct) operator-adapted frame thresholding approach which we show to provide a convergent regularization method with linear convergence rates. These results will be compared to the well-known analysis and synthesis variants of sparse $\ell^1$-regularization which are usually implemented thorough iterative schemes. If the frame is a basis (non-redundant case), the three versions of sparse regularization, namely synthesis and analysis variants of $\ell^1$ regularization as well as the DFD thresholding are equivalent. However, in the redundant case, those three approaches are pairwise different.

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Jürgen Frikel, Markus Haltmeier. 2019-12-12. Sparse regularization of inverse problems by operator-adapted frame thresholding. https://arxiv.org/abs/1909.09364

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