  This paper investigates the recovery of a spectrally sparse signal from its
partially revealed noisy entries within the framework of spectral compressive
sensing. Nonconvex optimization approaches have recently been proposed based on
low-rank Hankel matrix completion and projected gradient descent (PGD). The PGD
however involves unknown tuning parameters and its theoretical analysis is
available only in the absence of noise. In this paper, we propose a
hyperparameter-free, vanilla gradient descent (VGD) algorithm and prove that
the VGD enables robust recovery of an $N$-dimensional $K$-spectrally-sparse
signal from order $K^2 log^2N$ number of noisy samples under coherence and
other mild conditions. The above sample complexity increases by factor $logN$
as compared with PGD without noise. Numerical simulations are provided that
corroborate our analysis and show advantageous performances of VGD.
