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Sparse representation learning has recently gained a great success in signaland image processing, thanks to recent advances in dictionary learning. To thisend, the $\ell_0$-norm is often used to control the sparsity level.Nevertheless, optimization problems based on the $\ell_0$-norm are non-convexand NP-hard. For these reasons, relaxation techniques have been attracting muchattention of researchers, by priorly targeting approximation solutions (e.g.$\ell_1$-norm, pursuit strategies). On the contrary, this paper considers theexact $\ell_0$-norm optimization problem and proves that it can be solvedeffectively, despite of its complexity. The proposed method reformulates theproblem as a Mixed-Integer Quadratic Program (MIQP) and gets the global optimalsolution by applying existing optimization software. Because the maindifficulty of this approach is its computational time, two techniques areintroduced that improve the computational speed. Finally, our method is appliedto image denoising which shows its feasibility and relevance compared to thestate-of-the-art.

Une v\'eritable approche $\ell_0$ pour l'apprentissage de dictionnaire