In this paper, we study the spectral compressed sensing problem of recovering a spectrally sparse signal from linear compressive measurements. Exploiting the low-rank Hankel structure associated with the signal, we formulate a nonconvex fixed-rank optimization problem with a Hankel regularization. We propose a Superfast Iterative Hard Thresholding (SIHT) algorithm, where each iteration consists of a tangent-space least-squares update followed by a fixed-rank hard thresholding projection. SIHT is the first nonconvex algorithm with provable quadratic convergence for spectral compressed sensing. For random Gaussian measurements, it achieves exact recovery in the noiseless case and stable recovery in the noisy case with near-optimal sample complexity. To efficiently implement SIHT, we develop a parameterized conjugate gradient method for solving the least-squares subproblem, endowing SIHT with state-of-the-art computational and memory efficiency. Numerical experiments are provided to validate our results.

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