Geometrical properties and accelerated gradient solvers of non-convex phase retrieval

We consider recovering a signal x ∈ ℝn from the magnitudes of Gaussian measurements by minimizing a second order yet non-smooth loss function. By exploiting existing concentration results of the loss function, we show that the non-convex loss function satisfies several quadratic geometrical properties. Based on these geometrical properties, we characterize the linear convergence of the sequence of function graph generated by the gradient flow on minimizing the loss function. Furthermore, we propose an accelerated version of the gradient flow, and establish an in-exact linear convergence of the generated sequence of function graph by exploiting the quadratic geometries of the loss function. Then, we verify the numerical advantages of the proposed algorithms over other state-of-art algorithms.