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Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation

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by Yonina C. Eldar , Holger Rauhut
Citations:98 - 18 self
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BibTeX

@MISC{Eldar_averagecase,
    author = {Yonina C. Eldar and Holger Rauhut},
    title = { Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation},
    year = {}
}

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Abstract

In this paper, we consider recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relaxation based on a mixed matrix norm. Typically, worst-case analysis is carried out in order to analyze conditions under which the algorithms are able to recover any jointly sparse set of vectors. However, such an approach is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Previous work considered an average case analysis of thresholding and SOMP by imposing a probability model on the measured signals. In this paper, our main focus is on analysis of convex relaxation techniques. In particular, we focus on the mixed ℓ2,1 approach to multichannel recovery. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. Our probability bounds are valid and meaningful even for a small number of signals. Using the tools we develop to analyze the convex relaxation method, we also tighten the previous bounds for thresholding and SOMP.

Keyphrases

average case analysis    convex relaxation technique    incomplete measurement    standard sparse reconstruction method    sparse set    recovery failure    worst-case analysis    several approach    previous work    probability bound    convex relaxation    simultaneous orthogonal matching pursuit    single channel method    dictionary characteristic    mixed matrix norm    unknown sparse vector    previous bound    sparse multichannel signal    convex relaxation method    small number    probability model    main focus    joint sparse recovery    sparse recovery    mild condition   

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