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. In addition, we explore the Compatibility Condition and the Restricted Eigenvalue Condition in the noisy setup, as well as properties of neighbourly random polytopes.
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Lecué et al. (2017) studied this question.