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An Improved Estimate in the Restricted Isometry Problem

Part of the Lecture Notes in Mathematics book series (LNM,volume 2116)

Abstract

It is shown that for the n × n-Hadamard matrix (or, more generally, a bounded orthogonal matrix) the RIP-property for r-space vectors holds, with row restriction to a set S of size

$$\displaystyle{\vert S\vert < C(\varepsilon )(\log n)^{2}(\log r)r.}$$

This bound represents a slight improvement over (Rudelson and Vershynin, Commun Pure Appl Math 61:1025–1045, 2008) in that the power of the logarithm is decreased by one unit.

Keywords

  • Stochastic Process
  • Functional Analysis
  • Probability Theory
  • Probabilistic Statement
  • Discrete Geometry

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References

  1. M. Rudelson, R. Vershynin, On sparse reconstruction from Fourier and Gaussian measurements. Commun. Pure Appl. Math. 61(8), 1025–1045 (2008)

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Acknowledgements

Research supported in part by NSF Grant DMS 1301619.

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Correspondence to Jean Bourgain .

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© 2014 Springer International Publishing Switzerland

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Bourgain, J. (2014). An Improved Estimate in the Restricted Isometry Problem. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_5

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