A Preconditioned Hessian Proximal Algorithm for Spectral Compressed Sensing

We study the spectral compressed sensing (SCS) problem to recover spectrally sparse signals from their partially observed entries. The SCS problem can be reformulated as a low-rank Hankel matrix completion problem. Conventional first-order methods exhibit slow convergence rates, and fast convergence speed comes with sacrificing the accuracy of reconstruction. Specifically, a small regularization coefficient for the Hankel structure constraint is employed. To pursue fast convergence and accurate reconstruction, we propose a Hessian proximal gradient (HPG) method to select a larger regularization coefficient without compromising the reconstruction accuracy, and the computational complexity of the forward step with a preconditioned Hessian is reduced from
$O(n^{3})$
to
$O(\overline{r}^{2}n+rn1^{-}\text{og}n)$
for n-size r-spectrally-sparse signals. Numerical simulations demonstrate that HPG outperforms state-of-the-art approaches in terms of computational speed and reconstruction accuracy.