Provable Phase Retrieval with Mirror Descent
This addresses the phase retrieval problem in signal processing and optics, offering a provable method with theoretical guarantees, though it is incremental as it builds on existing mirror descent techniques.
The paper tackles the phase retrieval problem by proposing a mirror descent algorithm that removes the global Lipschitz continuity requirement, showing that with enough Gaussian measurements, it recovers the vector up to a sign change with high probability and exhibits local linear convergence with a dimension-independent rate.
In this paper, we consider the problem of phase retrieval, which consists of recovering an $n$-dimensional real vector from the magnitude of its $m$ linear measurements. We propose a mirror descent (or Bregman gradient descent) algorithm based on a wisely chosen Bregman divergence, hence allowing to remove the classical global Lipschitz continuity requirement on the gradient of the non-convex phase retrieval objective to be minimized. We apply the mirror descent for two random measurements: the \iid standard Gaussian and those obtained by multiple structured illuminations through Coded Diffraction Patterns (CDP). For the Gaussian case, we show that when the number of measurements $m$ is large enough, then with high probability, for almost all initializers, the algorithm recovers the original vector up to a global sign change. For both measurements, the mirror descent exhibits a local linear convergence behaviour with a dimension-independent convergence rate. Our theoretical results are finally illustrated with various numerical experiments, including an application to the reconstruction of images in precision optics.