Phase Transition in the One-bit Johnson-Lindenstrauss Lemma

The Johnson-Lindenstrauss Lemma (J-L Lemma) is a cornerstone of dimension reduction techniques. We study it in the one-bit context, namely we consider the unit sphere $ \mathbb S ^{N-1}$, with normalized geodesic metric, and map a finite set $ \mathbf{X} \subset \mathbb{S}^{N-1}$ into the Hamming cube $\mathbb{H}_m = \{0,1\}^m$, with normalized Hamming metric. We find that for $ 0< \delta <1$, and $m>\frac{\ln n}{2\delta^2}$ there is a $\delta$-RIP from $\mathbf{X}$ into $\mathbb{H}_m$. This is surprising as the value of $ m$ is virtually identical to best known bound linear J-L Lemma. In both the linear and one-bit case, the maps are randomly constructed. We show that the probability of $B_m$ being a $\delta$-RIP satisfies a phase transition. It passes from probability of nearly zero to nearly one with a very small change in $m$. Our proof relies on delicate properties of Bernoulli random variables.