Pré-Publication, Document De Travail Année : 2025

Geometry of Sparsity-Inducing Norms

Résumé

Sparse optimization seeks an optimal solution with few nonzero entries. To achieve this, it is common to add to the criterion a penalty term proportional to the $\ell_1$-norm, which is recognized as the archetype of sparsity-inducing norms. In this approach, the number of nonzero entries is not controlled a priori. By contrast, in this paper, we focus on finding an optimal solution with at most~$k$ nonzero coordinates (or for short, $k$-sparse vectors), where $k$ is a given sparsity level (or ``sparsity budget''). For this purpose, we study the class of generalized $k$-support norms that arise from a given source norm. When added as a penalty term, we provide conditions under which such generalized $k$-support norms promote $k$-sparse solutions. The result follows from an analysis of the exposed faces of closed convex sets generated by $k$-sparse vectors, and of how primal support identification can be deduced from dual information. Finally, we study some of the geometric properties of the unit balls for the $k$-support norms and their dual norms when the source norm belongs to the family of $\ell_p$-norms.
Fichier principal
Vignette du fichier
main_preprint_v1.pdf (600.31 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04886852 , version 1 (14-01-2025)

Identifiants

  • HAL Id : hal-04886852 , version 1

Citer

Jean-Philippe Chancelier, Michel de Lara, Antoine Deza, Lionel Pournin. Geometry of Sparsity-Inducing Norms. 2025. ⟨hal-04886852⟩
0 Consultations
0 Téléchargements

Partager

More