Hidden cliques and the certification of the restricted isometry property

Pascal Koiran (1, 2, 3) , Anastasios Zouzias (4)

1 LIP - Laboratoire de l'Informatique du Parallélisme
2 DCS - Department of Computer Science [University of Toronto]
3 MC2 - Modèles de calcul, Complexité, Combinatoire
4 Department of Computer Science

Compressed sensing is a technique for finding sparse solutions to underdetermined linear systems. This technique relies on properties of the sensing matrix such as the restricted isometry property. Sensing matrices that satisfy this property with optimal parameters are mainly obtained via probabilistic arguments. Deciding whether a given matrix satisfies the restricted isometry property is a non-trivial computational problem. Indeed, we show in this paper that restricted isometry parameters cannot be approximated in polynomial time within any constant factor under the assumption that the hidden clique problem is hard. Moreover, on the positive side we propose an improvement on the brute-force enumeration algorithm for checking the restricted isometry property.

Mots clés: compressed sensing; computational complexity; restricted isometry property; hidden clique problem

Domaines: Complexité [cs.CC]

Pascal Koiran, Anastasios Zouzias. Hidden cliques and the certification of the restricted isometry property. IEEE Transactions on Information Theory, 2014, 60 (8), pp.4999-5007. ⟨ensl-00747908⟩