The Johnson–Lindenstrauss Lemma Almost Characterizes Hilbert Space, But Not Quite

• Authors: William B. Johnson;Assaf Naor
• Venue: Discrete & Computational Geometry
• Year: 2010

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,…,x n ∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that ‖x i −x j ‖≤‖L(x i )−L(x j )‖≤O(1)⋅‖x i −x j ‖ for all i,j∈{1,…,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 ⊆Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.