Sparsest representations and approximations of an underdetermined linear system

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

1 ToxAlim - ToxAlim

In an underdetermined 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 sufficient and "almost" necessary condition to recover a sparsest representation with the basis pursuit. Unfortunately, this property cannot be easily checked. On the other hand, the mutual coherence is an easily checkable sufficient condition insuring the basis pursuit to recover one of the sparsest representations. Because the mutual coherence condition is too strong, it is hardly met in practice. Even if one of these conditions holds, 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]

Patrick J C Tardivel, Rémi Servien, Didier Concordet. Sparsest representations and approximations of an underdetermined linear system. 2018. ⟨hal-01535608v3⟩