Séminaire IMAGE : « Phase Retrieval with Bregman-based Geometry », Jean-Jacques Godem

Nous aurons le plaisir d’écouter Jean-Jacques Godem, doctorant de l’équipe IMAGE du GREYC, qui vient de soutenir sa thèse.
Il donnera un séminaire IMAGE, le jeudi 11 juillet 2024, à 14h00, en salle de séminaire F-200.

Titre: « Phase Retrieval with Bregman-based Geometry »

Résumé: 
In this work, we investigate the phase retrieval problem of real-valued signals in finite dimension, a challenge encountered across various scientific and engineering disciplines. It explores two complementary approaches: retrieval with and without regularization. In both settings, our work is focused on relaxing the Lipschitz-smoothness assumption generally required by first-order splitting algorithms, and which is not valid for phase retrieval cast as a minimization problem. The key idea here is to replace the Euclidean geometry by a non-Euclidean Bregman divergence associated to an appropriate kernel. We use a Bregman gradient/mirror descent algorithm with this divergence to solve the phase retrieval problem without regularization, and we show exact (up to a global sign) recovery both in a deterministic setting and with high probability for a sufficient number of random measurements (Gaussian and Coded Diffraction Patterns). Furthermore, we establish the robustness of this approach against small additive noise. Shifting to regularized phase retrieval, we first develop and analyze an Inertial Bregman Proximal Gradient algorithm for minimizing the sum of two functions in finite dimension, one of which is convex and possibly nonsmooth and the second is relatively smooth in the Bregman geometry. We provide both global and local convergence guarantees for this algorithm. Finally, we study noiseless and stable recovery of low complexity regularized phase retrieval. For this, we formulate the problem as the minimization of an objective functional involving a nonconvex smooth data fidelity term and a convex regularizer promoting solutions conforming to some notion of low-complexity related to their nonsmoothness points. We establish conditions for exact and stable recovery and provide sample complexity bounds for random measurements to ensure that these conditions hold. These sample bounds depend on the low complexity of the signals to be recovered. Our new results allow to go far beyond the case of sparse phase retrieval.

Keywords: phase retrieval, inverse problems, stability to noise, inertial Bregman proximal gradient, partly smooth function, trap avoidance, variational regularization, sparsity, exact recovery, low complexity prior, robustness.