Adaptive Signal Recovery by Convex Optimization

Mots clé :

• statistical signal processing
• harmonic recovery
• first-order proximal methods

We consider the problem of denoising a signal observed in Gaussian noise. In this problem, classical linear estimators are quasi-optimal provided that the set of possible signals is convex, compact, and known a priori. However, when the set is unspecified, designing an estimator which does not "know" the underlying structure of a signal yet has favorable theoretical guarantees of statistical performance remains a challenging problem. In this thesis, we study a new family of estimators for statistical recovery of signals satisfying certain time-invariance properties. Such signals are characterized by their harmonic structure, which is usually unknown in practice. We propose new estimators which are capable of exploiting the unknown harmonic structure of a signal to reconstruct. We demonstrate that these estimators admit theoretical performance guarantees, in the form of oracle inequalities, in a variety of settings. We provide efficient algorithmic implementation of these estimators via first-order optimization algorithms with non-Euclidean geometry, and evaluate them on synthetic data, as well as some real-world signals and images.

Directeurs:

• Mr Zaïd Harchaoui (PROFESSEUR - Université de Washington )
• Mr Anatoli Iouditski (Professeur - Université Grenoble Alpes )
• Mr Laurent Desbat (Professeur - Université Grenoble Alpes )

Raporteurs:

• Mr Olivier Cappé (CNRS - CNRS )
• Mr Arnak Dalalyan (Professeur - ENSAE ParisTech )

Examinateurs:

• Mr Yuri Golubev (PROFESSEUR - Université Aix-Marseille )
• Mme Céline Levy-Leduc (Professeure - AgroParisTech )
• Mr Gabriel Peyré (Directeur de Recherche - CNRS )