Generalized Restricted Isometry Property for alpha-stable random projections

The Restricted Isometry Property (RIP) is an important concept in compressed sensing. It is well known that many random matrices satisfy the RIP with high probability, whenever the entries of the random matrix have finite second order moment. Recent work in compressed sensing has shown that it is possible to do dimensionality reduction and signal reconstruction using Cauchy random projections. This suggests that the l1 distance is preserved when one projects a set of data points from a high-dimensional space, to one of lower dimension with a random matrix which does not have finite variance. This paper generalizes this concept where it is shown that α-stable projections, which preserve the lα distance, also satisfy a generalized RIP property and consequently reconstruction from α-stable projections is feasible.