A Coordinate Descent Approach to Atomic Norm Denoising

Abstract

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 O(N log N ), rendering the method efficient for large-scale problems. Through simulations, the proposed solver is shown to be faster than alternating direction method of multiplier (ADMM) or customized interior point SDP solver if the problems are sparse. It is demonstrated that the coordinate descent solver can be modified for AST with multiple dimensions and multiple measurement vectors as well as a variety of general basis.

arXiv comment14 pages, 12 figures, manuscript currently under review

References in corpus (7)

• The solution path of the generalized lasso
• Newtonized Orthogonal Matching Pursuit: Frequency Estimation over the Continuum
• Harnessing Sparsity over the Continuum: Atomic Norm Minimization for Super Resolution
• Super-Resolution Channel Estimation for Arbitrary Arrays in Hybrid Millimeter-Wave Massive MIMO Systems
• Low-complexity optimization for Two-Dimensional Direction-of-arrival Estimation via Decoupled Atomic Norm Minimization
• Single Snapshot Super-Resolution DOA Estimation for Arbitrary Array Geometries
• Optimal convex lifted sparse phase retrieval and PCA with an atomic matrix norm regularizer