A Coordinate Descent Approach to Atomic Norm Minimization

Atomic norm minimization is of great interest in various applications of sparse signal processing including super-resolution line-spectral estimation and signal denoising. In practice, atomic norm minimization (ANM) is formulated as a semi-definite programming (SDP) which is generally hard to solve. This work introduces a low-complexity, matrix-free method for solving ANM. The method uses the framework of coordinate descent and exploits the sparsity-induced nature of atomic-norm regularization. Specifically, an equivalent, non-convex formulation of ANM is first proposed. It is then proved that applying the coordinate descent framework on the non-convex formulation leads to convergence to the global optimal point. For the case of a single measurement vector of length N in discrete fourier transform (DFT) basis, the complexity of each iteration in the coordinate descent procedure is O(N log N ), rendering the proposed method efficient even for large-scale problems. The proposed coordinate descent framework can be readily modified to solve a variety of ANM problems, including multi-dimensional ANM with multiple measurement vectors. It is easy to implement and can essentially be applied to any atomic sets as long as a corresponding rank-1 problem can be solved. Through extensive numerical simulations, it is verified that for solving sparse problems the proposed method is much faster than the alternating direction method of multipliers (ADMM) or the customized interior point SDP solver.