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 paper, 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^N of S points and a distortion level ϵ > 0, as soon as M > M₀ = O(ϵ^-2 log S), we can (randomly) construct a mapping from (S, ℓ₂) to (δℤ^M, ℓ₁) that approximately preserves the pairwise distances between the points of S. Interestingly, compared with 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((log S/M)^1/2) when M increases. Moreover, for coarse quantization, i.e., for high δ compared with the set radius, the distortion is mainly additive, while for small δ we tend to a Lipschitz isometric embedding. Finally, we prove the existence of a nearly quasi-isometric embedding of (S, ℓ₂) into (δℤ^M, ℓ₂). This one involves a non-linear distortion of the ℓ₂-distance in S that vanishes for distant points in this set. Noticeably, the additive distortion in this case is slower, and decays as O((log S/M)^1/4).