Atoms of All Channels, Unite! Average Case Analysis of Multi-Channel Sparse Recovery Using Greedy Algorithms
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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.
KeywordsGreedy algorithms OMP Thresholding Multi-channel Average analysis
Mathematics Subject Classification (2000)41A28 41A46 60D05
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- 1.Baraniuk, R., Davenport, M., DeVore, R., Wakin, M.: A simple proof of the restricted isometry property for random matrices. Constr. Approx. (to appear) Google Scholar
- 2.Baron, D., Duarte, M., Sarvotham, S., Wakin, M., Baraniuk, R.: An information-theoretic approach to distributed compressed sensing. In: Proc. 45rd Conference on Communication, Control, and Computing (2005) Google Scholar
- 3.Baron, D., Wakin, M., Duarte, M., Sarvotham, S., Baraniuk, R.: Distributed compressed sensing. Preprint (2005) Google Scholar
- 6.Chen, J., Huo, X.: Sparse representations for multiple measurement vectors (MMV) in an over-complete dictionary. In: International Conference on Acoustics, Speech and Signal Processing (ICASSP-2005) (2005) Google Scholar
- 12.Gribonval, R., Nielsen, M.: Beyond sparsity: Recovering structured representations by l1 minimization and greedy algorithms. Publication interne 1684, IRISA, Rennes (2005) Google Scholar
- 13.Gribonval, R., Nielsen, M., Vandergheynst, P.: Towards an adaptive computational strategy for sparse signal approximation. Preprint of the Institut de Recherche en Informatique et Systèmes Aléatoires (IRISA) (2006) Google Scholar
- 14.Grimmett, G.R., Stirzaker, D.R.: Probability and Random Processes. Oxford University Press, London (2001) Google Scholar
- 18.Rauhut, H.: Stability results for random sampling of sparse trigonometric polynomials. IEEE Trans. Inf. Theory (to appear) Google Scholar
- 20.Rudelson, M., Vershynin, R.: Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements. In: Proc. CISS 2006 (40th Annual Conference on Information Sciences and Systems) (2006) Google Scholar
- 23.Taubman, D., Marcellin, W.: JPEG2000: Image Compression Fundamentals, Standards, and Practice. Springer, Berlin (2002) Google Scholar
- 25.Tropp, J.: Topics in sparse approximation. Ph.D. Thesis, University of Texas at Austin (2004) Google Scholar