# The origin of the restricted isometry property (RIP)

I have been looking for the origin of the restricted isometry property (RIP). Many papers cite the origin of the RIP in the following paper

but I cannot understand where the definition came from. Some others papers say that the RIP is very similar to the Johnson-Lindenstrauss lemma (JL), and in fact it is, but I am not sure that that it is the origin of the RIP. However, I haven't been able to find this origin, mathematically.

On the other hand, how can the RIP be proved with certain matrices? Let's say for example Gaussian matrices or a Fourier matrix?

• To address your 2nd question. Proving that a matrix satisfies the RIP is an NP hard problem. You would have to enumerate the possibilities and test them. For a Gaussian matrix, it can be shown that it satisfies the RIP with a high probability, – David Feb 19 '19 at 13:06
• @David can you explain me how it can be shown that a Gaussian Matrix satisfies the RIP with high probability. You can send me the paper if you are able and want to. – Lord Feb 19 '19 at 13:26
• The reference I have is: R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simpleproof of the restricted isometry property for random matrices,” Constructive Approximation, vol. 28, no. 3, pp. 253–263, 2008.Link here – David Feb 19 '19 at 16:35