Phase retrieval from STFT measurements via non-convex optimization

The problem of recovering a signal from its phaseless short-time Fourier transform (STFT) measurements arises in several applications, such as ultra-short pulse measurements and ptychography. The redundancy offered by the STFT enables unique recovery under mild conditions. We show that in some cases, the principle eigenvector of a designed matrix recovers the underlying signal. This matrix is constructed as the solution of a simple least-squares problem. When these conditions are not met, we suggest to use this principle eigenvector to initialize a gradient algorithm, minimizing a non-convex loss function. We prove that under appropriate conditions, this initialization results in a good estimate of the underlying signal. We further analyze the geometry of the loss function and show empirically that the gradient algorithm is robust to noise. Our method is both efficient and enjoys theoretical guarantees.