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.