Sparse recovery under weak moment assumptions
  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. 
Guillaume Lecu\'e and Shahar Mendelson
math.ST math.PR stat.TH