Smoothed Robust Phase Retrieval
This addresses a critical issue in scientific applications like imaging by providing theoretical guarantees for robust phase retrieval, though it is incremental as it builds on existing methods with smoothing.
The paper tackles the phase retrieval problem with infrequent but arbitrary corruptions by introducing smoothed robust phase retrieval (SRPR), proving it has no spurious local solutions under noiseless conditions and a benign landscape with corruptions, and demonstrating local linear convergence of gradient descent.
The phase retrieval problem in the presence of noise aims to recover the signal vector of interest from a set of quadratic measurements with infrequent but arbitrary corruptions, and it plays an important role in many scientific applications. However, the essential geometric structure of the nonconvex robust phase retrieval based on the $\ell_1$-loss is largely unknown to study spurious local solutions, even under the ideal noiseless setting, and its intrinsic nonsmooth nature also impacts the efficiency of optimization algorithms. This paper introduces the smoothed robust phase retrieval (SRPR) based on a family of convolution-type smoothed loss functions. Theoretically, we prove that the SRPR enjoys a benign geometric structure with high probability: (1) under the noiseless situation, the SRPR has no spurious local solutions, and the target signals are global solutions, and (2) under the infrequent but arbitrary corruptions, we characterize the stationary points of the SRPR and prove its benign landscape, which is the first landscape analysis of phase retrieval with corruption in the literature. Moreover, we prove the local linear convergence rate of gradient descent for solving the SRPR under the noiseless situation. Experiments on both simulated datasets and image recovery are provided to demonstrate the numerical performance of the SRPR.