## Low Complexity Regularization of Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. We consider regularization priors which promote solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds.

### Axis

- Recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification.
- Sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation.
- Convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem.

### Collaborators

Charles Deledalle |
IMB, Université Bordeaux 1, France |

Charles Dossal |
IMB, Université Bordeaux 1, France |

Jalal Fadili |
GREY'C, ENSICAEN, France |

Mohammad Golbabaee |
DSP, Rice University, USA |

Gabriel Peyré |
CEREMADE, Université Paris Dauphine, France |

Joseph Salmon |
TSI, Télécom ParisTech, France |

More information about my Ph. D. thesis.