On Data Increasing in Phase Retrieval via Quadratic Inversion: Flattening Manifold and Local Minima

Abstract

In this article, we address the question of traps in phase retrieval via quadratic inversion. First, we clarify the genesis of local minima from a geometrical perspective by showing how the shape of the manifold of data and the position of the data point are related to the presence of local minima in the objective functional. Later, we recall a mathematical condition for the lack of traps. Afterward, we illustrate how the ratio between the dimension of data space $M$ and the number of real unknowns $N_{r}$ impacts on the manifold of data and, consequently, on the presence of local minima. In particular, we show that if the dimension of data space is high enough, the manifold of data is sufficiently large and no traps appear in the functional. In the last part of this article, with reference to a particular unknown sequence, we establish the minimum value of the ratio ${M}/{N_{r}}$ such that no traps should appear in the functional. In this condition, phase retrieval via quadratic inversion exhibits a global convergence behavior, consequently, the actual solution of the problem can be reached regardless of the initial guess which can be chosen completely at random.