Strong Convexity of Affine Phase Retrieval

Affine phase retrieval refers to the process of recovering a signal from intensity measurements, with some entries known in advance. In this paper, we demonstrate that a natural least squares formulation for affine phase retrieval is strongly convex on the complete space under certain mild conditions, given that the measurement vectors are complex Gaussian random vectors and that the number of measurements is
$m\geq O(d\log d)$
, where
$d$
is the dimension of the 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 highlight a fundamental difference between affine phase retrieval and classical phase retrieval, where the least squares formulations for classical phase retrieval are non-convex.