Inexact alternating optimization for phase retrieval in the presence of outliers

a demo for robust phase retrieval using AltGD

C. Qian, X. Fu, N. D. Sidiropoulos, L. Huang and J. Xie, "Inexact alternating optimization for phase retrieval in the presence of outliers," IEEE Transactions on Signal Processing, vol. 65, no. 22, pp. 6069-6082, 2017.
Phase retrieval has been mainly considered in the
presence of Gaussian noise. However, the performance of the algorithms
proposed under the Gaussian noise model severely degrades
when grossly corrupted data, i.e., outliers, exist. This paper investigates
techniques for phase retrieval in the presence of heavy-tailed
noise, which is considered a better model for situations where outliers
exist. An p-norm (0 <p< 2) based estimator is proposed
for fending against such noise, and two-block inexact alternating
optimization is proposed as the algorithmic framework to tackle
the resulting optimization problem. Two specific algorithms are
devised by exploring different local approximations within this
framework. Interestingly, the core conditional minimization steps
can be interpreted as iteratively reweighted least squares and gradient
descent. Convergence properties of the algorithms are discussed,
and the Cramer–Rao bound (CRB) is derived. Simulations ´
demonstrate that the proposed algorithms approach the CRB and
outperform state-of-the-art algorithms in heavy-tailed noise.