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A Non-convex Approach to Non-negative Super-resolution: Theory and Algorithm

This paper considers the problem of super-resolution reconstruction by casting it as an optimization problem with positive constraints and non-convex objective function. Enforcing the solution to be simultaneously sparse and non-negative naturally leads to a non-convex l1/2 quasinorm minimization problem. A reweighted l1 norm minimization algorithm is proposed to solve this problem, which is tailored for l1/2 quasinorm minimization using the idea of Majorization-Minimization. Although the problem is non-convex and non-smooth, and the measurement matrix does not satisfy restricted isometry conditions, we are able to obtain deterministic stable reconstruction guarantees in presence of bounded noise by using the structure of the measurement matrix and non-negativity of the signal. Numerical results demonstrate that l1/2 minimization promotes sparser solution and outperforms l1 minimization.