Exact continuous relaxations of ℓ0-regularized criteria with non-quadratic data terms - Essafri, Mhamed; Calatroni, Luca; Soubies, Emmanuel (Journal of global optimization, 2025)
We consider the minimization of ℓ0-regularized criteria involving non-quadratic data terms such as the Kullback-Leibler divergence and the logistic regression, possibly combined with an ℓ2 regularization. We first prove the existence of global minimizers for such problems and characterize their local minimizers. Then, we propose a new class of continuous relaxations of the ℓ0 pseudo-norm, termed as ℓ0 Bregman Relaxations (B-rex). They are defined in terms of suitable Bregman distances and lead to exact continuous relaxations of the original ℓ0-regularized problem in the sense that they do not alter its set of global minimizers and reduce its non-convexity by eliminating certain local minimizers. Both features make such relaxed problems more amenable to be solved by standard non-convex optimization algorithms. In this spirit, we consider the proximal gradient algorithm and provide explicit computation of proximal points for the B-rex penalty in several cases. Finally, we report a set of numerical results illustrating the geometrical behavior of the proposed B-rex penalty for different choices of the underlying Bregman distance, its relation with convex envelopes, as well as its exact relaxation properties in 1D/2D and higher dimensions.