LANM: Learned Atomic Norm Minimization for Superfast Gridless Spectral Compressed Sensing

Atomic norm minimization (ANM) is a well-established approach to spectral compressed sensing that estimates the frequencies on the continuum and offers theoretical recovery guarantees. But its practical application is limited by the high computational cost associated with solving a semidefinite program. In this paper, we propose a novel matrix factorization formulation for ANM and develop a gradient descent algorithm to solve it with low per-iteration computational complexity. To further reduce the number of iterations, we propose a superfast deep learning method to implement the algorithm based on the deep unfolding technique, referred to as Learned ANM (LANM). LANM consists of only a few iterations, with each iteration efficiently computed using fast Fourier transforms (FFTs). Numerical experiments are provided to show that LANM achieves superior performance in both speed and reconstruction accuracy.