Sensitivity computation of the ℓ1 minimization problem and its application to dictionary design of ill-posed problems

Abstract

The ℓ1 minimization problem has been studied extensively in the past few years. Recently, there has been a growing interest in its application for inverse problems. Most studies have concentrated in devising ways for sparse representation of a solution using a given prototype dictionary. Very few studies have addressed the more challenging problem of optimal dictionary construction, and even these were primarily devoted to the simplistic sparse coding application. In this paper, sensitivity analysis of the inverse solution with respect to the dictionary is presented. This analysis reveals some of the salient features and intrinsic difficulties which are associated with the dictionary design problem. Equipped with these insights, we propose an optimization strategy that alleviates these hurdles while utilizing the derived sensitivity relations for the design of a locally optimal dictionary. Our optimality criterion is based on local minimization of the Bayesian risk, given a set of training models. We present a mathematical formulation and an algorithmic framework to achieve this goal. The proposed framework offers the design of dictionaries for inverse problems that incorporate non-trivial, non-injective observation operators, where the data and the recovered parameters may reside in different spaces. We test our algorithm and show that it yields improved dictionaries for a diverse set of inverse problems in geophysics and medical imaging.