Uniform Cramér–Rao Lower Bound for Phase Retrieval: An Empirical Study

This paper derives and analyzes the uniform Cramér–Rao Lower Bound (UCRLB) for the phase retrieval problem, and then compares the bias-variance trade-offs of several phase retrieval algorithms (e.g., Wirtinger flow (WF), Gerchberg-Saxton (GS), phaselift, majorize-minimize (MM), alternating direction method of multipliers (ADMM)) that were derived from maximum likelihood (ML) estimates where the measurements follow independent Gaussian distribution. We also consider regularizers that exploit assumed properties of the latent signal, e.g.,
$\ell_{2}$
norm and
$\ell_{1}$
norm (approximated by the Huber function) that corresponds to the sparsity of finite differences (anisotropic total variation (TV)) or of the detail coefficients of a discrete wavelet transform. Simulation results show that the classical CRLB fails to lower-bound the variances of phase retrieval algorithms, whereas the U CRLB provides a variance lower-bound. Simulation results also show that the regularized algorithms that better approximate the properties of the true signal have better bias-variance trade-offs (when compared to UCRLB).