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 \( 0< \delta \frac{\ln n}{2\delta^2}\) 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.