A remark on dimensionality reduction in discrete subgroups

TL;DR

This paper extends the Johnson-Lindenstrauss lemma to discrete subgroups, showing that high-dimensional point sets in integer lattices can be embedded into lower-dimensional lattices with controlled distortion.

Contribution

It proves a version of the Johnson-Lindenstrauss lemma for point sets in discrete subgroups, providing bounds on embedding dimension and distortion.

Findings

Existence of low-distortion embeddings into lower-dimensional lattices.

Embedding dimension scales as O(1/ε^2 log d).

Distortion is at most (1+ε+ε/(λλ₀)).

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 $d,\lambda_0,N_0\in\mathbb{N}$ and $\epsilon\in \left(0,\frac{1}{2}\right)$ suitably chosen, we show there exists a natural number $k=k(d,\epsilon)=O\left(\frac{1}{\epsilon^2}\log d\right)$, such that for every sufficiently large scaling factor $\lambda\in\mathbb{N}$ and any point set $\mathcal{D}\subset\frac{\lambda}{\lambda_0}\mathbb{Z}^d\cap B(0,\lambda N_0)$ with cardinality $d$, there exists an embedding $F:\mathcal{D}\to\frac{1}{\lambda_0}\mathbb{Z}^k$, with distortion at most $\left(1+\epsilon+\frac{\epsilon}{\lambda\lambda_0}\right)$.