Projected Wirtinger Gradient Descent for Low-Rank Hankel Matrix Completion in Spectral Compressed Sensing
This work addresses signal recovery in spectral compressed sensing, offering an incremental improvement for applications like communications or imaging by enhancing computational efficiency for large dimensions.
The paper tackles the problem of reconstructing spectrally sparse signals from limited time-domain samples by converting it into a low-rank Hankel matrix completion task, proposing a projected Wirtinger gradient descent (PWGD) algorithm that efficiently handles large-scale problems with demonstrated numerical efficiency.
This paper considers reconstructing a spectrally sparse signal from a small number of randomly observed time-domain samples. The signal of interest is a linear combination of complex sinusoids at $R$ distinct frequencies. The frequencies can assume any continuous values in the normalized frequency domain $[0,1)$. After converting the spectrally sparse signal recovery into a low rank structured matrix completion problem, we propose an efficient feasible point approach, named projected Wirtinger gradient descent (PWGD) algorithm, to efficiently solve this structured matrix completion problem. We further accelerate our proposed algorithm by a scheme inspired by FISTA. We give the convergence analysis of our proposed algorithms. Extensive numerical experiments are provided to illustrate the efficiency of our proposed algorithm. Different from earlier approaches, our algorithm can solve problems of very large dimensions very efficiently.