Low Complexity Regularizations of Inverse Problems

Defended on July 10, 2014. [HAL]


Francis Bach INRIA, École Normale Supérieure Member
Emmanuel Candès Stanford University Reviewer
Antonin Chambolle CNRS, École Polytechnique Member
Albert Cohen Université Pierre et Marie Curie President
Charles Dossal Université Bordeaux 1 Guest
Jalal Fadili ENSICAEN Member
Rémi Gribonval INRIA Rennes Reviewer
Gabriel Peyré CNRS, Université Paris-Dauphine Advisor
Vincent Rivoirard Université Paris-Dauphine Member


This thesis is concerned with recovery guarantees and sensitivity analysis of variational regularization for noisy linear inverse problems. This is cast as a convex optimization problem by combining a data fidelity and a regularizing functional promoting solutions conforming to some notion of low complexity related to their non-smoothness points. Our approach, based on partial smoothness, handles a variety of regularizers including analysis/structured sparsity, antisparsity and low-rank structure. We first give an analysis of the noise robustness guarantees, both in terms of the distance of the recovered solutions to the original object, as well as the stability of the promoted model space. We then turn to sensivity analysis of these optimization problems to observation perturbations. With random observations, we build unbiased estimator of the risk which provides a parameter selection scheme.

Keywords: inverse problem, variational regularization, low complexity prior, sparsity, robustness, sensitivity, risk estimation, degrees of freedom, parameter selection, partly smooth function.