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# Sparse recovery under weak moment assumptions

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### Abstract

We prove that iid random vectors that satisfy a rather weak moment assumption can be used as measurement vectors in Compressed Sensing, and the number of measurements required for exact reconstruction is the same as the best possible estimate -- exhibited by a random gaussian matrix. We also prove that this moment condition is necessary, up to a $$\log \log$$ factor. Applications to the Compatibility Condition and the Restricted Eigenvalue Condition in the noisy setup and to properties of neighbourly random polytopes are also discussed.

### Author and article information

###### Journal
2014-01-09
2015-03-03
1401.2188

math.ST math.PR stat.TH

Probability, Statistics theory