Breaking the k2 barrier for explicit RIP matrices

We give a new explicit construction of n x N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε>0, 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.

[1]  Jean Bourgain,et al.  On a variant of sum-product estimates and explicit exponential sum bounds in prime fields , 2009, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  Анатолий Алексеевич Карацуба,et al.  Двойные суммы Клоостермана@@@Double Kloosterman sums , 1999 .

[3]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[4]  Ronald A. DeVore,et al.  Deterministic constructions of compressed sensing matrices , 2007, J. Complex..

[5]  R. DeVore,et al.  A Simple Proof of the Restricted Isometry Property for Random Matrices , 2008 .

[6]  Stephen J. Dilworth,et al.  Explicit constructions of RIP matrices and related problems , 2010, ArXiv.

[7]  Vladimir N. Temlyakov,et al.  On the size of incoherent systems , 2011, J. Approx. Theory.

[8]  Deanna Needell,et al.  Uniform Uncertainty Principle and Signal Recovery via Regularized Orthogonal Matching Pursuit , 2007, Found. Comput. Math..

[9]  E. Livshitz.  On efficiency of Orthogonal Matching Pursuit , 2010, 1004.3946.

[10]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

[11]  Michael Rosen,et al.  A classical introduction to modern number theory , 1982, Graduate texts in mathematics.

[12]  Piotr Indyk,et al.  Combining geometry and combinatorics: A unified approach to sparse signal recovery , 2008, 2008 46th Annual Allerton Conference on Communication, Control, and Computing.

[13]  S. Mendelson,et al.  Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis , 2007 .

[14]  Shamgar Gurevich,et al.  Statistical RIP and Semi-Circle Distribution of Incoherent Dictionaries , 2009, ArXiv.

[15]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .

[16]  Entao Liu,et al.  Orthogonal Super Greedy Algorithm and Applications in Compressed Sensing ∗ , 2010 .

[17]  Michael Elad,et al.  On Lebesgue-type inequalities for greedy approximation , 2007, J. Approx. Theory.

[18]  Jean Bourgain,et al.  Multilinear Exponential Sums in Prime Fields Under Optimal Entropy Condition on the Sources , 2009 .

[19]  Terence Tao,et al.  Additive combinatorics , 2007, Cambridge studies in advanced mathematics.

[20]  W. B. Johnson,et al.  Extensions of Lipschitz mappings into Hilbert space , 1984 .

[21]  E. D. Gluskin.  The octahedron is badly approximated by random subspaces , 1986 .

[22]  Douglas R. Woodall.  A theorem on cubes , 1977 .

[23]  J. Bourgain,et al.  Exponential sum estimates over a subgroup in an arbitrary finite field , 2011 .

[24]  M. Rudelson,et al.  On sparse reconstruction from Fourier and Gaussian measurements , 2008 .

[25]  S. Muthukrishnan,et al.  Approximation of functions over redundant dictionaries using coherence , 2003, SODA '03.

[26]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[27]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[28]  Béla Bollobás,et al.  Sums in the grid , 1996, Discret. Math..