Accelerated Recovery of Spectrally Sparse Signals Viamodified Proximal Gradient in Hankel Space

Abstract

We study the reconstruction of an n-size r-spectrally-sparse signal from its m-size subset, with a specific focus on addressing the challenges arising from low sampling density. This recovery task can be transformed into a low-rank Hankel matrix completion problem in array form. Existing approaches show slow convergence when the sampling ratio p=m/n is low. To address this issue, we propose a nonconvex method composed of a Hankel matrix projection and a modified proximal gradient (PG) algorithm named HPPG. By preserving the Hankel structure, HPPG enables a larger step size for gradient descent as p decreases, resulting in accelerated convergence. Additionally, our approach reduces the computational complexity per iteration from O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) to O(rnlogn+r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> n) by leveraging the structured Hankel matrix. Numerical results illustrate that HPPG surpasses state-of-the-art methods in terms of both computational efficiency and reconstruction accuracy.