Sparse regularization of inverse problems by operator-adapted frame thresholding

TL;DR

This paper introduces a direct, operator-adapted frame thresholding method for sparse regularization of inverse problems, generalizing SVD, with proven convergence and linear rates, and compares it to traditional iterative methods.

Contribution

It proposes a novel non-iterative regularization approach based on diagonal frame decomposition, extending sparse regularization techniques to redundant frames with proven convergence.

Findings

The DFD thresholding method converges with linear rates.

In the basis case, all three regularization methods are equivalent.

In the redundant case, the methods differ significantly.

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.