Provable Phase retrieval via Mirror descent
Jean-Jacques Godeme  1@  , Jalal Fadili  1  , Xavier Buet  2  , Myriam Zerrad  2  , Claude Amra  2  , Michel Lequime  2  
Normandie Univ, ENSICAEN, CNRS, GREYC, France
2 : Institut Fresnel
Aix-Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France

We consider the problem of phase retrieval, 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 through the use of a non-euclidean Bregman divergence, to solve the nonconvex
formulation of the phase retrieval problem. We show that when the measurements are sufficiently large,
with high probability we can recover the desired vector up to a global sign change. Our set-up uses careful
initialization via a spectral method and refines it using the mirror descent with a backtracking procedure to find
the optimal solution. We show local linear convergence with a rate and step-size independent of the dimension.
Our results are stated for two types of measurements: those drawn independently from the standard Gaussian,
and those obtained by Coded Diffraction Patterns (CDP) for Randomized Fourier Transform.

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