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Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation
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@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
