Optimal terminal dimensionality reduction in Euclidean space

Abstract

Let ε ∈ (0, 1) and X ⊂ R d be arbitrary with |X| having size n > 1. The Johnson-Lindenstrauss lemma states there exists f :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 log n). Our proof is via a tighter analysis of (a specific instantiation of) the embedding recipe of [MMMR18]. * Harvard University. shyamnarayanan@college.harvard.edu. Supported by a PRISE fellowship and a Herchel-Smith Fellowship.† Harvard University. minilek@seas.harvard.edu.

Labelings vs. Embeddings: On Distributed Representations of Distances

“…, x n }. Narayanan and Nelson [NN18] (improving a previous result by Mahabadi et al [MMMR18]) constructed a terminal version of the JL transform: Specifically, given a set K of k points in 2 there is an embedding f of the entire…”

“…Since every p , p ∈ [1, 2], embeds isometrically into squared-L 2 (equivalently, its snowflake embeds into L 2 ), this implies a labeling with the same performance for p as well, see Theorem 3.1. Furthermore, we show in Theorem 3.1 (using [NN18]) that this labeling can be prioritized to achieve distortion 1 + with label size O( −2 log j).…”

Lossless Prioritized Embeddings

“…The constant loss in the first part of the scheme has to do with an outer extension, implicitly developed in [EFN17], and explicated in [MMMR18,NN19]. Bi-Lipschitz outer extensions have been a focus of recent research [MMMR18,NN19,EN18], where they were studied in the context of Johnson-Lindenstrauss dimension reduction [MMMR18,NN19], and in the context of doubling metrics [EN18]. In both these contexts it was shown that the loss can be made at most 1 + , for an arbitrarily small > 0.…”

Oblivious dimension reduction for k -means: beyond subspaces and the Johnson-Lindenstrauss lemma

“…Terminal embeddings for Euclidean spaces were studied in the work [30,46,50]. In particular, Mahabadi et al [46] showed that a target dimension of m ∈ O(ε −4 log n) was sufficient, which was very recently by Narayanan and Nelson [50] to m ∈ O(ε −2 log n), which is optimal.…”

“…) rows, this results in a target dimension of order m ∈ O(ε −4 log ε −1 log k), due to Theorem 1.1. of Narayanan and Nelson [50]. We then compute a coreset P in the embedded space.…”