On the Convergence of the Fast Griffin-Lim Algorithm

The Fast Griffin-Lim Algorithm (FGLA) is one of the most widely used iterative methods for phase reconstruction and signal estimation from modified short-time Fourier trans- forms. Unlike the classic Griffin-Lim Algorithm (GLA), however, the convergence of FGLA has not yet been fully proved, with existing theoretical guarantees holding only for momenta signif- icantly smaller than those found to perform best in practice. In this letter, we build upon the appendix of the original paper by Griffin and Lim and formulate GLA as a gradient descent algorithm. From this, we show that FGLA corresponds to an accelerated gradient descent method with constant momentum. We then derive a sufficient condition ensuring convergence over a broad range of momenta and formulate a criterion to assess its validity in numerical experiments, which we find is typically satisfied within the first few iterations.