  In this paper, we aim to reconstruct an n-dimensional real vector from m
phaseless measurements corrupted by an additive noise. We extend the noiseless
framework developed in [15], based on mirror descent (or Bregman gradient
descent), to deal with noisy measurements and prove that the procedure is
stable to (small enough) additive noise. In the deterministic case, we show
that mirror descent converges to a critical point of the phase retrieval
problem, and if the algorithm is well initialized and the noise is small
enough, the critical point is near the true vector up to a global sign change.
When the measurements are i.i.d Gaussian and the signal-to-noise ratio is large
enough, we provide global convergence guarantees that ensure that with high
probability, mirror descent converges to a global minimizer near the true
vector (up to a global sign change), as soon as the number of measurements m is
large enough. The sample complexity bound can be improved if a spectral method
is used to provide a good initial guess. We complement our theoretical study
with several numerical results showing that mirror descent is both a
computationally and statistically efficient scheme to solve the phase retrieval
problem.
