"Accelerating FGLA using convex optimization"Nenov, RossenThe Griffin-Lim algorithm (GLA) is a famous method to solve the phase retrieval problem in the time-frequency setting. It aims to solve two related but different problems respectively, namely constructing a signal from a valid spectrogram with no phase information (Phase recovery) and constructing a signal from a synthetic or modified Short-Time Fourier transform (STFT) magnitude. These problems can be seen as an optimization task. We try to find elements in the intersection the convex set C_1, the range of the analysis operator of a frame, and the non-convex set C_2, i.e. those coefficients having the fixed magnitude. The GLA projects the iterates alternating onto C_1 and onto C_2 and the convergence of this method is known. Convex optimization provides fast algorithms for finding elements in the intersection of two convex sets, which resulted in the Fast Griffin-Lim algorithm (FGLA). In this approach we will use the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) directly. By defining the algorithm and the initial values suitably, we can write the projection onto C_2 as the projection onto its convex hull and therefore the fast convergence properties of FISTA to the global minimum hold, whereas for the GLA only the convergence to critical points, which are not necessarily the global minimum, is known. |
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