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

TL;DR

This paper presents an exact approach to $ ext{l}_0$-norm based dictionary learning by reformulating it as a MIQP, achieving global optimal solutions and demonstrating effectiveness in image denoising.

Contribution

It introduces a novel method that solves the $ ext{l}_0$-norm optimization exactly using MIQP, with techniques to improve computational efficiency.

Findings

Effective global optimal solutions for $ ext{l}_0$-norm problems.

Improved computational speed techniques.

Successful application to image denoising.

Abstract

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