A Coordinate Descent Approach to Atomic Norm Denoising

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 semi-definite programming (SDP) that is generally hard to solve. This work introduces a low-complexity solver for a type of ANM known as atomic norm soft thresholding (AST). The proposed method uses the framework of coordinate descent and exploits the sparsity-inducing nature of atomic norm regularization. Specifically, this work first provides an equivalent, non-convex formulation of AST. It is then proved that applying a coordinate descent algorithm on the non-convex formulation leads to convergence to the global solution. For the case of a single measurement vector of length $N$ and complex exponential basis, the complexity of each step in the coordinate descent procedure is $\mathcal{O}(N\log N)$ , rendering the method efficient for large-scale problems. Through simulations, for sparse problems the proposed solver is shown to be faster than alternating direction method of multiplier (ADMM) or customized interior point SDP solver. Numerical simulations demonstrate that the coordinate descent solver can be modified for AST with multiple dimensions and multiple measurement vectors as well as a variety of other continuous basis.