New and Improved Johnson-Lindenstrauss Embeddings via the Restricted Isometry Property

Consider anm×N matrix Φ with the Restricted Isometry Property of order k and level δ, that is, the norm of any k-sparse vector in R is preserved to within a multiplicative factor of 1±δ under application of Φ. We show that by randomizing the column signs of such a matrix Φ, the resulting map with high probability embeds any fixed set of p = O(e) points in R into R without distorting the norm of any point in the set by more than a factor of 1± 4δ. 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 & δ−4 log(p) log(N) appearing in Ailon and Liberty [3] to m & δ−2 log(p) log(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.