Spectral Super-Resolution on the Unit Circle Via Gradient Descent

We study the spectral super-resolution problem, which concerns the construction of an undamped spectrally sparse signal and its frequencies from its partially revealed entries. We propose a nonconvex method composed of a Hankel-Toeplitz matrix factorization model and a gradient descent algorithm termed as HT-GD. The model is equivalent to an ℓ₀ norm con-strained problem, which ensures that the all signal structures including the spectral poles lying on the unit circle are exploited. The gradient descent algorithm, consisting of spectral initialization and iterative refinement, is computationally efficient. Numerical results demonstrate that our method out-performs state-of-the-art approaches in terms of accuracy and computational speed.