Breaking the k2 barrier for explicit RIP matrices

Authors:
Jean Bourgain;Stephen J. Dilworth;Kevin Ford;Sergei V. Konyagin;Denka Kutzarova
Affiliations:
Institute For Advanced Study, Princeton, NJ, USA;University of South Carolina, Columbia, SC, USA;University of Illinois, Urbana, IL, USA;Steklov Mathematical Institute, Moscow, Russian Fed.;Bulgarian Academy of Sciences, Sofia, Bulgaria
Venue:
Proceedings of the forty-third annual ACM symposium on Theory of computing
Year:
2011

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Abstract

We give a new explicit construction of n x N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε0, large k and k2-ε ≤ N ≤ k2+ε, we construct RIP matrices of order k with n=O(k2-ε). This overcomes the natural barrier n k2 for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure.