15 July 2011 Explicit constructions of RIP matrices and related problems
Jean Bourgain, Stephen Dilworth, Kevin Ford, Sergei Konyagin, Denka Kutzarova
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Duke Math. J. 159(1): 145-185 (15 July 2011). DOI: 10.1215/00127094-1384809


We give a new explicit construction of n×N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ϵ>0, large N, and any n satisfying N1ϵnN, we construct RIP matrices of order kn1/2+ϵ and constant δ=nϵ. This overcomes the natural barrier k=O(n1/2) 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. We also give a construction of sets of n complex numbers whose kth moments are uniformly small for 1kN (Turán's power sum problem), which improves upon known explicit constructions when (logN)1+o(1)n(logN)4+o(1). This latter construction produces elementary explicit examples of n×N matrices that satisfy the RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (logN)1+o(1)n(logN)5/2+o(1).


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Jean Bourgain. Stephen Dilworth. Kevin Ford. Sergei Konyagin. Denka Kutzarova. "Explicit constructions of RIP matrices and related problems." Duke Math. J. 159 (1) 145 - 185, 15 July 2011. https://doi.org/10.1215/00127094-1384809


Published: 15 July 2011
First available in Project Euclid: 11 July 2011

zbMATH: 1236.94027
MathSciNet: MR2817651
Digital Object Identifier: 10.1215/00127094-1384809

Primary: 11T23
Secondary: 11B13 , 11B30 , 41A46 , 94A12 , 94B60

Rights: Copyright © 2011 Duke University Press


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Vol.159 • No. 1 • 15 July 2011
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