The Generalized Likelihood Ratio Test and the Sparse Representations Approach

  • Jean Jacques Fuchs
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6134)


When sparse representation techniques are used to tentatively recover the true sparse underlying model hidden in an observation vector, they can be seen as solving a joint detection and estimation problem. We consider the ℓ2 − ℓ1 regularized criterion, that is probably the most used in the sparse representation community, and show that, from a detection point of view, minimizing this criterion is similar to applying the Generalized Likelihood Ratio Test. More specifically tuning the regularization parameter in the criterion amounts to set the threshold in the Generalized Likelihood Ratio Test.


False Alarm Discrete Fourier Transform Sparse Representation Observation Vector Generalize Likelihood Ratio Test 
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  1. 1.
    Donoho, D.: Compresssed sensing. IEEE-T-IT 52, 1289–1306 (2006)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bofil, P., Zibulevski, M.: Underdetermined blind source separation using sparse representations. In: Signal Processing, November 2001, vol. 81, pp. 2353–2362. Elsevier, Amsterdam (2001)Google Scholar
  3. 3.
    Tibshirani, R.: Regression shrinkage and selection via the Lasso. J. Royal Statist., Soc. B 58(1), 267–288 (1996)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Kwakernaak, H.: Estimation of pulse heights and arrival times. Automatica 16, 367–377 (1980)zbMATHCrossRefGoogle Scholar
  5. 5.
    Fuchs, J.J.: On the Application of the Global Matched Filter to DOA Estimation with Uniform Circular Arrays. IEEE-T-SP 49, 702–709 (2001)CrossRefGoogle Scholar
  6. 6.
    Van Trees, H.L.: Detection, Modulation and Estimation Theory, part 1. John Wiley and Sons, Chichester (1968)Google Scholar
  7. 7.
    Cox, D., Hinkley, D.: Theoretical Statistics. Chapman and Hall, Boca Raton (1974)zbMATHGoogle Scholar
  8. 8.
    Fuchs, J.J.: Detection and estimation of superimposed signals. In: Proc. IEEE ICASSP, Seattle, vol. III, pp. 1649–1652 (1998)Google Scholar
  9. 9.
    Fletcher, R.: Practical Methods of Optimization. John Wiley and Sons, Chichester (1987)zbMATHGoogle Scholar
  10. 10.
    Scharf, L.L.: Statistical signal processing, detection, estimation and time series analysis. Addison Wesley, Reading (1991)zbMATHGoogle Scholar
  11. 11.
    Hotelling, H.: Tubes and spheres in n-spaces, and a class of statistical problems. Amer. J. Math. 61, 440–460 (1939)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Weyl, H.: On the volume of tubes. Amer. J. Math. 61, 461–472 (1939)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Fuchs, J.J.: On sparse representations in arbitrary redundant basis. IEEE-T-IT 50(6), 1341–1344 (2004)CrossRefGoogle Scholar
  14. 14.
    Villier, E., Vezzosi, G.: Caractéristiques opérationnelles de réception d’une classe de détecteurs. Gretsi, 15ème colloque, Juan les Pins, 125–128 (September 1995)Google Scholar
  15. 15.
    Fuchs, J.J.: A sparse representation criterion: recovery conditions and implementation issues. In: 17th World Congress IFAC, Seoul, July 2008, pp. 12425–12429 (2008)Google Scholar
  16. 16.
    Porat, B.: Digital Processing of Random Signals. Theory and Methods. Prentice Hall, New Jersey (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jean Jacques Fuchs
    • 1
  1. 1.IRISAUniv. de Rennes 1Rennes CedexFrance

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