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The “Traps” Issue in a Non Linear Inverse Problem: The Phase Retrieval in Circular Case

In this paper we address the question of the traps (local minima, quasi flat regions, and saddle points with null gradient) in phase recovery method based on the minimization of a quartic functional. In particular, we focus on the key role played by the ratio between the “essential dimension of data” Mind and the number of real unknowns Nr and we illustrate how it impacts on the presence of traps in the functional to be minimized. First we show that such functional flattens when the essential dimension of data increases; and later, with reference to two different unknown sequences, we find the minimum value of the essential dimension of data such that no traps exist in the functional. In this condition the recovery of the solution is ensured independently of the initial guess that can be chosen completely at random.