Instance Optimality and Quotient Property

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


This chapter introduces the notion of instance optimality. While uniform 1-instance optimality is a practical concept in compressive sensing, uniform 2-instance optimality is shown not to be. A new property, called quotient property, is then developed to analyze measurement–reconstruction schemes. This property of the measurement matrix, coupled with equality-constrained 1-minimization, guarantees robustness of the scheme under measurement errors as well as nonuniform 2-instance optimality. The quotient property is proved to hold with high probability for Gaussian matrices and, modulo a slight modification, for subgaussian matrices.


uniform instance optimality gaussian matrices subgaussian matrices stability robustness 1-minimization quotient property nonuniform instance optimality 


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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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