Greedy Solution of Ill-Posed Problems: Error Bounds and Exact Inversion

TL;DR

This paper investigates the use of orthogonal matching pursuit (OMP) for solving ill-posed inverse problems, providing error bounds and conditions for exact support recovery in noisy settings, with applications to mass spectrometry and digital holography.

Contribution

The paper develops new theoretical results for OMP in ill-posed inverse problems, extending recovery guarantees beyond traditional incoherence conditions.

Findings

OMP can achieve exact support recovery in ill-posed inverse problems.

Derived a priori estimates for practical experimental setup verification.

First analysis of resolution power in digital droplet holography.

Abstract

The orthogonal matching pursuit (OMP) is an algorithm to solve sparse approximation problems. Sufficient conditions for exact recovery are known with and without noise. In this paper we investigate the applicability of the OMP for the solution of ill-posed inverse problems in general and in particular for two deconvolution examples from mass spectrometry and digital holography respectively. In sparse approximation problems one often has to deal with the problem of redundancy of a dictionary, i.e. the atoms are not linearly independent. However, one expects them to be approximatively orthogonal and this is quantified by the so-called incoherence. This idea cannot be transfered to ill-posed inverse problems since here the atoms are typically far from orthogonal: The ill-posedness of the operator causes that the correlation of two distinct atoms probably gets huge, i.e. that two atoms can look much alike. Therefore one needs conditions which take the structure of the problem into account and work without the concept of coherence. In this paper we develop results for exact recovery of the support of noisy signals. In the two examples in mass spectrometry and digital holography we show that our results lead to practically relevant estimates such that one may check a priori if the experimental setup guarantees exact deconvolution with OMP. Especially in the example from digital holography our analysis may be regarded as a first step to calculate the resolution power of droplet holography.