A Low-Rank Projected Proximal Gradient Method for Spectral Compressed Sensing

This paper presents a new approach to the recovery of spectrally sparse signals (SSS) from partially observed entries, addressing challenges posed by large-scale data and heavy-noise environments. The SSS reconstruction can be formulated as a non-convex low-rank Hankel recovery problem. Traditional formulations for SSS recovery often suffer from reconstruction inaccuracies due to unequally weighted norms and over-relaxation of the Hankel structure in noisy conditions. Additionally, a critical limitation of standard proximal gradient (PG) methods for solving this optimization problem is their slow convergence. We overcome this issue by introducing a more accurate formulation and proposing the Low-rank Projected Proximal Gradient (LPPG) method, designed to efficiently converge to stationary points through a two-step process. The first step involves a modified PG approach, allowing for a constant step size independent of the signal size, significantly accelerating the gradient descent phase. The second step employs a subspace projection strategy, optimizing within a low-rank matrix space to further decrease the objective function. Both steps of the LPPG method are meticulously tailored to exploit the intrinsic low-rank and Hankel structures of the problem, thereby enhancing computational efficiency. Numerical simulations demonstrate substantial improvements in both efficiency and recovery accuracy of the LPPG method compared to existing benchmark algorithms. This performance gain is particularly pronounced in scenarios with significant noise, showcasing the method's robustness and applicability to large-scale SSS recovery tasks.