New and improved Johnson-Lindenstrauss embeddings via the Restricted Isometry Property

Consider an m by N matrix Phi with the Restricted Isometry Property of order k and level delta, that is, the norm of any k-sparse vector in R^N is preserved to within a multiplicative factor of 1 +- delta under application of Phi. We show that by randomizing the column signs of such a matrix Phi, the resulting map with high probability embeds any fixed set of p = O(e^k) points in R^N into R^m without distorting the norm of any point in the set by more than a factor of 1 +- delta. Consequently, matrices with the Restricted Isometry Property and with randomized column signs provide optimal Johnson-Lindenstrauss embeddings up to logarithmic factors in N. In particular, our results improve the best known bounds on the necessary embedding dimension m for a wide class of structured random matrices; for partial Fourier and partial Hadamard matrices, we improve the recent bound m = O(delta^-4 log(p) log^4(N)) appearing in Ailon and Liberty to m = O(delta^-2 log(p) log^4(N)), which is optimal up to the logarithmic factors in N. Our results also have a direct application in the area of compressed sensing for redundant dictionaries.