A Preconditioned Fast Iterative Hard Thresholding Algorithm for Spectrally Sparse Signal Reconstruction

Spectrally sparse signals arise from various applications, including seismic imaging, autoregression, and magnetic resonance spectroscopy. Reconstructing a spectrally sparse signal from sub-samples on the uniform grid in the time domain can be reformulated as a low-rank Hankel matrix completion problem. Although the fast iterative hard thresholding (FIHT) algorithm is an efficient non-convex solver for this problem, it relies on the canonical metric in the matrix space. To speed up FIHT, we introduce a novel algorithm, preconditioned fast iterative hard thresholding (PFIHT), which adopts a data-driven metric by employing preconditioning techniques. Numerical experiments demonstrate that PFIHT outperforms FIHT and projected gradient descent (PGD) in terms of the number of iterations required for convergence and overall computational time.