Nonlinear dimension reduction via outer Bi-Lipschitz extensions

Mahabadi, Sepideh; Makarychev, Konstantin; Makarychev, Yury; Razenshteyn, Ilya

doi:10.1145/3188745.3188828

Cited by 22 publications

(41 citation statements)

References 39 publications

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“…We outline both the approach of previous works as well as our own. In all these approaches, one starts with an embedding f : X → ℓ m 2 with good distortion, then defines an outer extension f Ext as introduced in [EFN17] and defined explicitly in [MMMR18].…”

Section: Overview Of Approachmentioning

confidence: 99%

“…In [EFN17,MMMR18] and the current work, a terminal embedding is obtained by, for each u ∈ R d \X, defining an outer extension f Ext act identically on X for any u, u ′ ∈ R d , we can then define our final terminal embedding bỹ…”

Section: Overview Of Approachmentioning

confidence: 99%

“…Remark. Unlike in the JL lemma for which the embedding f may be linear, the terminal embedding we analyze here (as was also in the case in [EFN17,MMMR18]) is nonlinear, as it must be. To see that it must be nonlinear, consider that any linear embedding with constant terminal distortion, x → Πx for some matrix Π ∈ R m×d , must have Π(x − y) = 0 for any x ∈ X and any y ∈ R d on the unit sphere centered at x.…”

Section: Introductionmentioning

confidence: 99%

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Optimal terminal dimensionality reduction in Euclidean space

Narayanan

Nelson

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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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 :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.

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Section: Overview Of Approachmentioning

confidence: 99%

Section: Overview Of Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal terminal dimensionality reduction in Euclidean space

Narayanan

Nelson

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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“…6 It is moreover open if these embeddings can be made prioritized. This question was explicitly studied in [MMMR18], where a prioritized version of JL dimension reduction was shown, albeit with a much larger (than 1 + ) distortion of O(log log j), and under a weaker notion of prioritized distortion than the one we study here. 7 While preparing this submission we were informed [FGK19] that a similar result was lately achieved independently of us by Filtser, Gottlieb and Krauthgamer.…”

Section: Introductionmentioning

confidence: 96%

“…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.…”

Section: Introductionmentioning

confidence: 99%