A Smoothing Stochastic Phase Retrieval Algorithm for Solving Random Quadratic Systems

A novel Stochastic Smoothing Phase Retrieval (SSPR) algorithm is studied to reconstruct an unknown ζsignal x ∈ ℝ
ⁿ
or ${{\mathbb{C}}^n}$ from a set of absolute square projections y
_k
= |〈a
_k
; x〉|
²
. This inverse problem is known in the literature as Phase Retrieval (PR). Recent works have shown that the PR problem can be solved by optimizing a nonconvex and non-smooth cost function. Contrary to the recent truncated gradient descend methods developed to solve the PR problem (using truncation parameters to bypass the non-smoothness of the cost function), the proposed algorithm approximates the cost function of interest by a smooth function. Optimizing this smooth function involves a single equation per iteration, which leads to a simple scalable and fast method especially for large sample sizes. Extensive simulations suggest that SSPR requires a reduced number of measurements for recovering the signal x, when compared to recently developed stochastic algorithms. Our experiments also demonstrate that SSPR is robust to the presence of additive noise and has a speed of convergence comparable with that of state-of-the-art algorithms.