  We consider a least absolute deviation (LAD) approach to the robust phase
retrieval problem that aims to recover a signal from its absolute measurements
corrupted with sparse noise. To solve the resulting non-convex optimization
problem, we propose a robust alternating minimization (Robust-AM) derived as an
unconstrained Gauss-Newton method. To solve the inner optimization arising in
each step of Robust-AM, we adopt two computationally efficient methods for
linear programs. We provide a non-asymptotic convergence analysis of these
practical algorithms for Robust-AM under the standard Gaussian measurement
assumption. These algorithms, when suitably initialized, are guaranteed to
converge linearly to the ground truth at an order-optimal sample complexity
with high probability while the support of sparse noise is arbitrarily fixed
and the sparsity level is no larger than $1/4$. Additionally, through
comprehensive numerical experiments on synthetic and image datasets, we show
that Robust-AM outperforms existing methods for robust phase retrieval offering
comparable theoretical performance
