The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite

TL;DR

This paper explores the relationship between the Johnson-Lindenstrauss lemma and the structure of normed spaces, showing that spaces satisfying the lemma are nearly Euclidean, but some can still have high Euclidean distortion.

Contribution

It demonstrates that spaces satisfying the J-L lemma are almost Euclidean, but also constructs examples with arbitrarily high Euclidean distortion, highlighting limits of the lemma's characterization.

Findings

Spaces satisfying the J-L lemma embed into Hilbert space with very low distortion.

Existence of normed spaces satisfying J-L but with high Euclidean distortion in some subspaces.

Almost Euclidean structure is closely related to the J-L lemma in normed spaces.

Abstract

Let $X$ be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer $n$ and any $x_1,\ldots,x_n\in X$ there exists a linear mapping $L:X\to F$, where $F\subseteq X$ is a linear subspace of dimension $O(\log n)$, such that $\|x_i-x_j\|\le\|L(x_i)-L(x_j)\|\le O(1)\cdot\|x_i-x_j\|$ for all $i,j\in \{1,\ldots, n\}$. We show that this implies that $X$ is almost Euclidean in the following sense: Every $n$-dimensional subspace of $X$ embeds into Hilbert space with distortion $2^{2^{O(\log^*n)}}$. On the other hand, we show that there exists a normed space $Y$ which satisfies the J-L lemma, but for every $n$ there exists an $n$-dimensional subspace $E_n\subseteq Y$ whose Euclidean distortion is at least $2^{\Omega(\alpha(n))}$, where $\alpha$ is the inverse Ackermann function.