A Quantized Johnson Lindenstrauss Lemma: The Finding of Buffon's Needle

In 1733, Georges-Louis Leclerc, Comte de Buffon in France, set the ground of geometric probability theory by defining an enlightening problem: What is the probability that a needle thrown randomly on a ground made of equispaced parallel strips lies on two of them? In this work, we show that the solution to this problem, and its generalization to N dimensions, allows us to discover a quantized form of the Johnson-Lindenstrauss (JL) Lemma, i.e., one that combines a linear dimensionality reduction procedure with a uniform quantization of precision δ > 0. In particular, given a finite set S ⊂ R of S points and a distortion level > 0, as soon as M > M0 = O( −2 logS), we can (randomly) construct a mapping from (S, `2) to ( (δ Z) , `1) that approximately preserves the pairwise distances between the points of S. Interestingly, compared to the common JL Lemma, the mapping is quasi-isometric and we observe both an additive and a multiplicative distortions on the embedded distances. These two distortions, however, decay as O( √ logS/M) when M increases. Moreover, for coarse quantization, i.e., for high δ compared to the set radius, the distortion is mainly additive, while for small δ we tend to a Lipschitz isometric embedding. Finally, we show that there exists “almost” a quasi-isometric embedding of (S, `2) in ((δZ) , `2). This one involves a non-linear distortion of the `2-distance in S that vanishes for distant points in this set. Noticeably, the additive distortion in this case is slower and decays as O((logS/M)).