Generalizing the Johnson-Lindenstrauss lemma to k-dimensional affine subspaces

• Autorzy: Alon Dmitriyuk , Yehoram Gordon
• Języki publikacji: EN

Abstrakty

EN

Let ε > 0 and 1 ≤ k ≤ n and let ${W_{l}}_{l=1}^{p}$ be affine subspaces of ℝⁿ, each of dimension at most k. Let $m = O(ε^{-2}(k + log p))$ if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map $H: ℝⁿ → ℝ^{m}$ such that for all 1 ≤ l ≤ p and $x,y ∈ W_{l}$ we have ||x-y||₂ ≤ ||H(x)-H(y)||₂ ≤ (1+ε)||x-y||₂, i.e. the distance distortion is at most 1 + ε. The estimate on m is tight in terms of k and p whenever ε < 1, and is tight on ε,k,p whenever ε ≥ 1. We extend these results to embeddings into general normed spaces Y.

Kategorie tematyczne

• 46B07: Local theory of Banach spaces
• 46B09: Probabilistic methods in Banach space theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2009
• Tom: 195
• Numer: 3
• Strony: 227-241

Daty

wydano

2009

Twórcy

autor

Alon Dmitriyuk

• Department of Mathematics, Technion, Haifa 32000, Israel

autor

Yehoram Gordon

• Department of Mathematics, Technion, Haifa 32000, Israel

Identyfikatory

DOI

10.4064/sm195-3-3