Accelerated Griffin-Lim Algorithm: A Fast and Provably Converging Numerical Method for Phase Retrieval

The recovery of a signal from the magnitudes of its transformation, like the Fourier transform, is known as the phase retrieval problem and is of big relevance in various fields of engineering and applied physics. In this paper, we present a fast inertial/momentum based algorithm for the phase retrieval problem. Our method can be seen as an extended algorithm of the Fast Griffin-Lim Algorithm, a method originally designed for phase retrieval in acoustics. The new numerical algorithm can be applied to a more general framework than acoustics, and as a main result of this paper, we prove a convergence guarantee of the new scheme. Consequently, we also provide an affirmative answer for the convergence of its ancestor Fast Griffin-Lim Algorithm, whose convergence remained unproven in the past decade. In the final chapter, we complement our theoretical findings with numerical experiments for the Short Time Fourier Transform phase retrieval and compare the new scheme with the Griffin-Lim Algorithm, the Fast Griffin-Lim Algorithm, and two other iterative algorithms typically used in acoustics.