  We consider the robust phase retrieval problem of recovering the unknown
signal from the magnitude-only measurements, where the measurements can be
contaminated by both sparse arbitrary corruption and bounded random noise. We
propose a new nonconvex algorithm for robust phase retrieval, namely Robust
Wirtinger Flow to jointly estimate the unknown signal and the sparse
corruption. We show that our proposed algorithm is guaranteed to converge
linearly to the unknown true signal up to a minimax optimal statistical
precision in such a challenging setting. Compared with existing robust phase
retrieval methods, we achieve an optimal sample complexity of $O(n)$ in both
noisy and noise-free settings. Thorough experiments on both synthetic and real
datasets corroborate our theory.
