Inverse Problem Regularization with Weak Decomposable Priors. Part I: Recovery Guarantees

This first talk is dedicated to assessing the theoretical recovery performance of this class of regularizers. We consider regularizations with convex positively 1homogenous functionals (in fact gauges) which obey a weak decomposability property. The weak decomposability will promote solutions of the inverse problem conforming to some notion of simplicity/low complexity by living on a low dimensional sub-space. This family of priors encompasses many special instances routinely used in regularized inverse problems such as l^1, l^1-l^2 (group sparsity), nuclear norm, or the l^\infty norm. The weak decomposability requirement is flexible enough to cope with analysis-type priors that include a pre-composition with a linear operator, such as for instance the total variation and polyhedral gauges. Weak decomposability is also stable under summation of regularizers, thus enabling to handle mixed regularizations.