Provable Phase retrieval via Mirror Descent

Jean-Jacques Godeme (1, 2, 3, 4) , Jalal Fadili (1, 2, 3, 4) , Xavier Buet (5, 6, 7) , Myriam Zerrad (5, 6, 7) , Michel Lequime (5, 6, 7) , Claude Amra (5, 6, 7)

1 Equipe Image - Laboratoire GREYC - UMR6072
2 NU - Normandie Université
3 ENSICAEN - École Nationale Supérieure d'Ingénieurs de Caen
4 GREYC - Groupe de Recherche en Informatique, Image et Instrumentation de Caen
5 FRESNEL - Institut FRESNEL
6 AMU - Aix Marseille Université
7 ECM - École Centrale de Marseille

We consider the problem of phase retrieval that consists in recovering an n-dimensional real vector from the magnitude of its m-linear measurements. This paper presents a new approach allowing to lift the classical global Lipschitz continuity requirement on the gradient of the non-convex objective to minimize. We propose a mirror descent algorithm based on a wisely chosen Bregman divergence. We show that when the number of measurements m is large enough, the mirror descent algorithm, carefully initialized, converges linearly with a dimension-independent convergence rate. Consequently, the original signal can be reconstructed exactly up to a global sign change. We state our results for two types of measurements: iid standard Gaussian and those obtained by Coded Diffraction Patterns (CDP) for Randomized Fourier Transform.

Domaines: Optimisation et contrôle [math.OC]

Jean-Jacques Godeme, Jalal Fadili, Xavier Buet, Myriam Zerrad, Michel Lequime, et al.. Provable Phase retrieval via Mirror Descent. SIAM Journal on Imaging Sciences, 2023, 16 (3), pp.1106. ⟨hal-03622580v4⟩