The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite Conference Paper uri icon

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, . . . , 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 2O(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 2 Ω(α(n)), where α is the inverse Ackermann function. Copyright © by SIAM.

author list (cited authors)

  • Johnson, W. B., Naor, A., & SIAM, A.

complete list of authors

  • Johnson, William B||Naor, Assaf||SIAM, ACM

publication date

  • September 2009