Optimal terminal dimensionality reduction in Euclidean space
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Let ε∈(0,1) and X⊂ R^d be arbitrary with |X| having size n>1. The Johnson-Lindenstrauss lemma states there exists f:X→ R^m with m = O(ε^-2 n) such that ∀ x∈ X ∀ y∈ X, x-y_2 <f(x)-f(y)_2 < (1+ε)x-y_2 . We show that a strictly stronger version of this statement holds, answering one of the main open questions of [MMMR18]: "∀ y∈ X" in the above statement may be replaced with "∀ y∈ R^d", so that f not only preserves distances within X, but also distances to X from the rest of space. Previously this stronger version was only known with the worse bound m = O(ε^-4 n). Our proof is via a tighter analysis of (a specific instantiation of) the embedding recipe of [MMMR18].
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