Sparsest representations and approximations of a high-dimensional linear system

Patrick J C Tardivel (1) , Rémi Servien (1) , Didier Concordet (1)

1 ToxAlim - ToxAlim

In a high-dimensional linear system of equations, constrained l1 minimization methods such as the basis pursuit or the lasso are often used to recover one of the sparsest representations or approximations of the system. The null space property is a sucient and "almost" necessary condition to recover a sparsest representation with the basis pursuit. Unfortunately, this property can not be easily checked. On the other hand, the mutual coherence or the restricted isometry property are checkable sucient conditions insuring the basis pursuit to recover one of the sparsest representation. Because both of these conditions are too strong, they are hardly met in practice. Even with these conditions, to our knowledge, there is no theoretical result insuring that the lasso solution is one of the sparsest approximations. In this article, we study a novel constrained problem that gives, without any condition, one of the sparsest representations or approximations. To solve this problem, we provide a numerical method and we prove its convergence. Numerical experiments show that this approach gives better results than both the basis pursuit problem and the reweighted l1 minimization problem.

Mots clés: Basis pursuit; Lasso; Sparsest representations; Sparsest approximations

Domaines: Statistiques [math.ST]; Mathématiques [math]; Sciences de l'ingénieur [physics]

Patrick J C Tardivel, Rémi Servien, Didier Concordet. Sparsest representations and approximations of a high-dimensional linear system. 2017. ⟨hal-01535608v1⟩