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We consider the problem of recovering a signal from the magnitudes of affinemeasurements, which is also known as {\em affine phase retrieval}. In thispaper, we formulate affine phase retrieval as an optimization problem anddevelop a second-order algorithm based on Newton method to solve it. Besidesbeing able to convert into a phase retrieval problem, affine phase retrievalhas its unique advantages in its solution. For example, the linear informationin the observation makes it possible to solve this problem with second-orderalgorithms under complex measurements. Another advantage is that our algorithmdoesn't have any special requirements for the initial point, while anappropriate initial value is essential for most non-convex phase retrievalalgorithms. Starting from zero, our algorithm generates iteration point byNewton method, and we prove that the algorithm can quadratically converge tothe true signal without any ambiguity for both Gaussian measurements and CDPmeasurements. In addition, we also use some numerical simulations to verify theconclusions and to show the effectiveness of the algorithm.

The Newton method for affine phase retrieval