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

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Dmitriyuk A., Gordon Y.

Studia Mathematica

2009

tom 195

nr 3

227-241

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.

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1 Studia Mathematica

1 Dmitriyuk A.

1 Gordon Y.

1 2009

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