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 transforms. 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 significantly 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.