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

Abstract

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 l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> quasinorm minimization problem. A reweighted l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm minimization algorithm is proposed to solve this problem, which is tailored for l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> 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 l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> minimization promotes sparser solution and outperforms l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> minimization.