IVDST: A Fast Algorithm for Atomic Norm Minimization in Line Spectral Estimation

This letter presents a fast algorithm named iterative Vandermonde decomposition and shrinkage-thresholding (IVDST), which offers a low-complexity solution to atomic norm minimization (ANM) in off-grid compressed sensing for line spectral estimation from few measurements. It implements the ANM principle via the accelerated proximal gradient (APG) technique, without invoking computationally expensive semidefinite programming (SDP). To approximate the proximal operator in each APG iteration, Vandermonde decomposition is applied to utilize the Toeplitz structure inherent in the line spectral model, and the low-rank property of the Toeplitz-structured matrix is enforced via a simple shrinkage-thresholding operation. The IVDST algorithm effectively reduces the order of computational complexity compared to SDP-based solutions. It also offers an explicit way to bridge the ANM principle with classic super-resolution line spectral estimation algorithms, such as MUSIC.