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

Abstract

Consider an $m \times N$ matrix $\Phi$ with the restricted isometry property of order k and level $\delta$; that is, the norm of any k-sparse vector in $\mathbb{R}^N$ is preserved to within a multiplicative factor of $1 \pm \delta$ 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 $\mathbb{R}^N$ into $\mathbb{R}^m$ without distorting the norm of any point in the set by more than a factor of $1 \pm 4 \delta$. 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 $m \gtrsim \delta^{-4} \log(p) \log^4(N)$ given by Ailon and Liberty to $m \gtrsim \delta^{-2} \log(p) \log^4(N)$, 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.