  The recovery of a signal from the intensity measurements with some entries
being known in advance is termed as {\em affine phase retrieval}. In this
paper, we prove that a natural least squares formulation for the affine phase
retrieval is strongly convex on the entire space under some mild conditions,
provided the measurements are complex Gaussian random vecotrs and the
measurement number $m \gtrsim d \log d$ where $d$ is the dimension of signals.
Based on the result, we prove that the simple gradient descent method for the
affine phase retrieval converges linearly to the target solution with high
probability from an arbitrary initial point. These results show an essential
difference between the affine phase retrieval and the classical phase
retrieval, where the least squares formulations for the classical phase
retrieval are non-convex.
