Sparse Solutions of Underdetermined Systems

  • Simon Foucart
  • Holger Rauhut
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

The notions of sparsity and compressibility are formally defined in this chapter, and some useful inequalities are established along the way. Then the question about the minimal number of linear equations to solve underdetermined systems admitting s-sparse solutions is answered. This is equivalent to the question about the minimal number of linear measurements to recover s-sparse vectors via \(\ell_{0}\)-minimization, which is shown to be NP-hard in general. A practical (but unstable) procedure to recover all s-sparse vectors using the minimal number of Fourier measurements is also presented.

Keywords

0-norm sparsity compressibility best s-term approximation nonincreasing rearrangement p-space weak p-space 0-minimization partial Fourier matrix Prony method NP-hardness exact cover by 3-sets problem 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Simon Foucart
    • 1
  • Holger Rauhut
    • 2
  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA
  2. 2.Lehrstuhl C für Mathematik (Analysis)RWTH Aachen UniversityAachenGermany

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