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Let ε > 0 and 1 ≤ k ≤ n and let {Wl}pl=1 be affine subspaces of R , 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 : R → R such that for all 1 ≤ l ≤ p and x, y ∈ Wl 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 .

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