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Journal of Fourier Analysis and Applications

, Volume 14, Issue 5–6, pp 655–687 | Cite as

Atoms of All Channels, Unite! Average Case Analysis of Multi-Channel Sparse Recovery Using Greedy Algorithms

  • Rémi Gribonval
  • Holger Rauhut
  • Karin SchnassEmail author
  • Pierre Vandergheynst
Article

Abstract

This paper provides new results on computing simultaneous sparse approximations of multichannel signals over redundant dictionaries using two greedy algorithms. The first one, p-thresholding, selects the S atoms that have the largest p-correlation while the second one, p-simultaneous matching pursuit (p-SOMP), is a generalisation of an algorithm studied by Tropp in (Signal Process. 86:572–588, 2006). We first provide exact recovery conditions as well as worst case analyses of all algorithms. The results, expressed using the standard cumulative coherence, are very reminiscent of the single channel case and, in particular, impose stringent restrictions on the dictionary.

We unlock the situation by performing an average case analysis of both algorithms. First, we set up a general probabilistic signal model in which the coefficients of the atoms are drawn at random from the standard Gaussian distribution. Second, we show that under this model, and with mild conditions on the coherence, the probability that p-thresholding and p-SOMP fail to recover the correct components is overwhelmingly small and gets smaller as the number of channels increases.

Furthermore, we analyse the influence of selecting the set of correct atoms at random. We show that, if the dictionary satisfies a uniform uncertainty principle (Candes and Tao, IEEE Trans. Inf. Theory, 52(12):5406–5425, 2006), the probability that simultaneous OMP fails to recover any sufficiently sparse set of atoms gets increasingly smaller as the number of channels increases.

Keywords

Greedy algorithms OMP Thresholding Multi-channel Average analysis 

Mathematics Subject Classification (2000)

41A28 41A46 60D05 

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Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  • Rémi Gribonval
    • 1
  • Holger Rauhut
    • 2
  • Karin Schnass
    • 3
    Email author
  • Pierre Vandergheynst
    • 3
  1. 1.Universitè Rennes I, IRISARennes CedexFrance
  2. 2.Hausdorff Center for MathematicsUniversity of BonnBonnGermany
  3. 3.Signal Processing Laboratories, LTS2Ecole Polytechnique Fédérale de Lausanne (EPFL)LausanneSwitzerland

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