Hidden cliques and the certification of the restricted isometry property

Abstract

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.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1103.4984

References in corpus (3)

• On Low Rank Matrix Approximations with Applications to Synthesis Problem in Compressed Sensing
• On Verifiable Sufficient Conditions for Sparse Signal Recovery via ${\ell }_1$ Minimization
• The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing

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