  Phase retrieval (PR) is an ill-conditioned inverse problem which can be found
in various science and engineering applications. Assuming sparse priority over
the signal of interest, recent algorithms have been developed to solve the
phase retrieval problem. Some examples include SparseAltMinPhase (SAMP), Sparse
Wirtinger flow (SWF) and Sparse Truncated Amplitude flow (SPARTA). However, the
optimization cost functions of the mentioned algorithms are non-convex and
non-smooth. In order to fix the non-smoothness of the cost function, the SPARTA
method uses truncation thresholds to calculate a truncated step update
direction. In practice, the truncation procedure requires calculating more
parameters to obtain a desired performance in the phase recovery. Therefore,
this paper proposes an algorithm called SPRSF (Sparse Phase retrieval via
Smoothing Function) to solve the sparse PR problem by introducing a smoothing
function. SPRSF is an iterative algorithm where the update step is obtained by
a hard thresholding over a gradient descent direction. Theoretical analyses
show that the smoothing function uniformly approximates the non-convex and
non-smooth sparse PR optimization problem. Moreover, SPRSF does not require the
truncation procedure used in SPARTA. Numerical tests demonstrate that SPRSF
performs better than state-of-the-art methods, especially when there is no
knowledge about the sparsity $k$. In particular, SPRSF attains a higher mean
recovery rate in comparison with SPARTA, SAMP and SWF methods, when the
sparsity varies for the real and complex cases. Further, in terms of the
sampling complexity, the SPRSF method outperforms its competitive alternatives.
