Constructive Approximation

, Volume 45, Issue 1, pp 113–127 | Cite as

Orthogonal Matching Pursuit Under the Restricted Isometry Property

  • Albert CohenEmail author
  • Wolfgang Dahmen
  • Ronald DeVore


This paper is concerned with the performance of orthogonal matching pursuit (OMP) algorithms applied to a dictionary \(\mathcal{D}\) in a Hilbert space \(\mathcal{H}\). Given an element \(f\in \mathcal{H}\), OMP generates a sequence of approximations \(f_n\), \(n=1,2,\ldots \), each of which is a linear combination of n dictionary elements chosen by a greedy criterion. It is studied whether the approximations \(f_n\) are in some sense comparable to best n-term approximation from the dictionary. One important result related to this question is a theorem of Zhang (IEEE Trans Inf Theory 57(9):6215–6221, 2011) in the context of sparse recovery of finite dimensional signals. This theorem shows that OMP exactly recovers n-sparse signals with at most An iterations, provided the dictionary \(\mathcal{D}\) satisfies a restricted isometry property (RIP) of order An for some constant A, and that the procedure is also stable in \(\ell ^2\) under measurement noise. The main contribution of the present paper is to give a structurally simpler proof of Zhang’s theorem, formulated in the general context of n-term approximation from a dictionary in arbitrary Hilbert spaces \(\mathcal{H}\). Namely, it is shown that OMP generates near best n-term approximations under a similar RIP condition.


Orthogonal matching pursuit Best n-term approximation Instance optimality Restricted isometry property 

Mathematics Subject Classification

94A12 94A15 68P30 41A46 15A52 


  1. 1.
    Candes, E., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59, 1207–1223 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cohen, A., Dahmen, W., DeVore, R.: Compressed sensing and best \(k\)-term approximation. J. Am. Math. Soc. 22, 211–231 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    DeVore, R.A.: Nonlinear approximation. Acta Numer. 7, 51–150 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    DeVore, R., Petrova, G., Wojtaszczyk, P.: Instance-optimality in probability with an \(\ell _1\)-minimization decoder. Appl. Comput. Harmon. Anal. 27(3), 275–288 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Foucart, S.: Stability and robustness of weak orthogonal matching pursuits. Springer Proc. Math. Stat. Recent Adv. Harmon. Anal. Appl. 25, 395–405 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Birkhäuser Verlag, Basel (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Livshitz, E.D.: On the optimality of the orthogonal greedy algorithm for \(\mu \)-coherent dictionaries. J. Approx. Theory 164–5, 668–681 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Livshitz, E.D., Temlyakov, V.N.: Sparse approximation and recovery by greedy algorithms. IEEE Trans. Inf. Theory 60, 3989–4000 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Temlyakov, V.N.: Greedy approximation. Acta Numer. 17, 235–409 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Temlyakov, V.N.: Greedy Approximation. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2011)Google Scholar
  11. 11.
    Temlyakov, V.N.: Sparse approximation and recovery by greedy algorithms in Banach spaces. Forum Math. Sigma (2014). doi: 10.1017/fms.2014.7 MathSciNetzbMATHGoogle Scholar
  12. 12.
    Tropp, J.A.: Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50, 2231–2242 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wang, J., Shim, B.: Exact recovery of sparse signals via orthogonal matching pursuit: How many iterations do we need?. arXiv:1211.4293v5 (2015)
  14. 14.
    Zhang, T.: Sparse recovery with orthogonal matching pursuit under RIP. IEEE Trans. Inf. Theory 57(9), 6215–6221 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie CurieParisFrance
  2. 2.Institut für Geometrie und Praktische MathematikRWTH AachenAachenGermany
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA

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