Robust Wirtinger Flow for Phase Retrieval with Arbitrary Corruption

We consider the 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 improved the statistical error rate by a factor of $\sqrt{n/m}$ where $n$ is the dimension of the signal and $m$ is the sample size, provided a refined characterization of the corruption fraction requirement, and relaxed the lower bound condition on the number of corruption. In the noise-free case, our algorithm converges to the unknown signal at a linear rate and achieves optimal sample complexity up to a logarithm factor. Thorough experiments on both synthetic and real datasets corroborate our theory.

Comments: 32 pages, 5 figures, 2 tables

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