A remark on dimensionality reduction in discrete subgroups

TL;DR

This work extends the Johnson-Lindenstrauss dimension reduction to discretized data lying in the fixed lattice $\frac{1}{\lambda_0}\mathbb{Z}^d$, proving there exists an embedding into $\frac{1}{\lambda_0}\mathbb{Z}^k$ with $k=O\left(\frac{1}{\varepsilon^2}\log d\right)$ that preserves distances up to $1+\varepsilon+\frac{\varepsilon}{\lambda\lambda_0}$ for sufficiently large scale $\lambda$. The approach combines a standard JL embedding with a rotation-based lattice alignment enabled by a Ziegler-type theorem, ensuring the rotated image lies near $\frac{1}{\lambda_0}\mathbb{Z}^k$ and yields a mapping into the lattice with bounded coordinates. A key technical ingredient is a rotation $\rho(\lambda)$ that makes the $\lambda$-scaled JL image arbitrarily close to the target lattice for large $\lambda$, enabling a precise lattice embedding with distinct images. The results provide a practical, bounded-coordinate dimension-reduction framework for discretized data, with explicit parameter relationships among $d$, $\lambda_0$, $N_0$, and $\varepsilon$.

Abstract

In this short note, we prove a version of the Johnson-Lindenstrauss flattening Lemma for point sets taking values in discrete subgroups. More precisely, given $d,λ_0,N_0\in\mathbb{N}$ and $ε\in \left(0,\frac{1}{2}\right)$ suitably chosen, we show there exists a natural number $k=k(d,ε)=O\left(\frac{1}{ε^2}\log d\right)$, such that for every sufficiently large scaling factor $λ\in\mathbb{N}$ and any point set $\mathcal{D}\subset\fracλ{λ_0}\mathbb{Z}^d\cap B(0,λN_0)$ with cardinality $d$, there exists an embedding $F:\mathcal{D}\to\frac{1}{λ_0}\mathbb{Z}^k$, with distortion at most $\left(1+ε+\fracε{λλ_0}\right)$.

Theorems & Definitions (5)

• Proposition 1.1: Main Proposition
• Theorem 1.2: Ziegler's Theorem
• Lemma 2.1
• proof
• proof : Proof of the \ref{['prop: JLdis']}