  Phase retrieval (PR) is an inverse problem about recovering a signal from
phaseless linear measurements. This problem can be effectively solved by
minimizing a nonconvex amplitude-based loss function. However, this loss
function is non-smooth. To address the non-smoothness, a series of methods have
been proposed by adding truncating, reweighting and smoothing operations to
adjust the gradient or the loss function and achieved better performance. But
these operations bring about extra rules and parameters that need to be
carefully designed. Unlike previous works, we present a smooth amplitude flow
method (SAF) which minimizes a novel loss function, without additionally
modifying the gradient or the loss function during gradient descending. Such a
new heuristic can be regarded as a smooth version of the original non-smooth
amplitude-based loss function. We prove that SAF can converge geometrically to
a global optimal point via the gradient algorithm with an elaborate
initialization stage with a high probability. Substantial numerical tests
empirically illustrate that the proposed heuristic is significantly superior to
the original amplitude-based loss function and SAF also outperforms other
state-of-the-art methods in terms of the recovery rate and the converging
speed. Specially, it is numerically shown that SAF can stably recover the
original signal when number of measurements is smaller than the
information-theoretic limit for both the real and the complex Gaussian models.
