Advertisement

Mathematical Notes

, Volume 104, Issue 5–6, pp 859–865 | Cite as

On the Recovery of an Integer Vector from Linear Measurements

  • S. V. KonyaginEmail author
Article
  • 9 Downloads

Abstract

Let 1 ≤ 2lm < d. A vector x ∈ ℤd is said to be l-sparse if it has at most l nonzero coordinates. Let an m × d matrix A be given. The problem of the recovery of an l-sparse vector x ∈ Zd from the vector y = Ax ∈ Rm is considered. In the case m = 2l, we obtain necessary conditions and sufficient conditions on the numbers m, d, and k ensuring the existence of an integer matrix A all of whose elements do not exceed k in absolute value which makes it possible to reconstruct l-sparse vectors in ℤd. For a fixed m, these conditions on d differ only by a logarithmic factor depending on k.

Keywords

nonsingular matrix lattices successive minima 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Davenport, M. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing (Cambridge Univ. Press, Cambridge, 2012), pp. 1–64.Google Scholar
  2. 2.
    M. Fornasier and H. Rauhut, “Compressive sensing,” in Handbook of Mathematical Methods in Imaging (Springer-Verlag, Berlin, 2011), pp. 187–228.CrossRefGoogle Scholar
  3. 3.
    L. Fukshansky, D. Needell, and B. Sudakov, An Algebraic Perspective on Integer Sparse Recovery, arXiv: 1801.01256 (2018).Google Scholar
  4. 4.
    J. Bourgain, V. H. Vu, and Ph. M. Wood, “On the singularity probability of discrete random matrices,” J. Funct. Anal. 258 (2), 559–603 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    I. M. Vinogradov, Foundations of Number Theory (Nauka, Moscow, 1981) [in Russian].zbMATHGoogle Scholar
  6. 6.
    M. Henk, “Successive minima and lattice points,” Rend. Circ. Mat. Palermo (2) Suppl. 70, 377–384 (2002).MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations