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.

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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.