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This paper investigates the phase retrieval problem, which aims to recover asignal from the magnitudes of its linear measurements. We develop statisticallyand computationally efficient algorithms for the situation when themeasurements are corrupted by sparse outliers that can take arbitrary values.We propose a novel approach to robustify the gradient descent algorithm byusing the sample median as a guide for pruning spurious samples ininitialization and local search. Adopting the Poisson loss and the reshapedquadratic loss respectively, we obtain two algorithms termed median-TWF andmedian-RWF, both of which provably recover the signal from a near-optimalnumber of measurements when the measurement vectors are composed of i.i.d.Gaussian entries, up to a logarithmic factor, even when a constant fraction ofthe measurements are adversarially corrupted. We further show that bothalgorithms are stable in the presence of additional dense bounded noise. Ouranalysis is accomplished by developing non-trivial concentration results ofmedian-related quantities, which may be of independent interest. We providenumerical experiments to demonstrate the effectiveness of our approach.

Median-Truncated Nonconvex Approach for Phase Retrieval with Outliers