A Continuous Relaxation of the Constrained $$\ell _2-\ell _0$$ Problem

We focus on the minimization of least square loss function under a k-sparse constraint encoded by $$\ell _0$$ pseudo-norm. This is non-convex, non-continuous and NP-hard problem. Recently, for penalized form (sum penalty term), relaxation has been introduced which strong results in terms minimizers. continuous does not change global minimizers, among other favorable properties. The question that driven this paper following: can problem be developed following same idea steps as _2-\ell problem? calculate convex envelope constrained when observation matrix orthogonal propose non-smooth, non-convex functional. give some equivalence minimizers between original relaxed problems. subgradient calculated well proximal operator new regularization term, we an algorithm ensures convergence to critical point apply single-molecule localization microscopy compare with well-known sparse schemes. proposed are good state-of-the-art form, while fixing constant usually more intuitive than parameter.