Nonlinear dimension reduction via outer Bi-Lipschitz extensions

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

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contribution

6 Scopus citations

Abstract

We introduce and study the notion of an outer bi-Lipschitz extension of a map between Euclidean spaces. The notion is a natural analogue of the notion of a Lipschitz extension of a Lipschitz map. We show that for every map f there exists an outer bi-Lipschitz extension f whose distortion is greater than that of f by at most a constant factor. This result can be seen as a counterpart of the classic Kirszbraun theorem for outer bi-Lipschitz extensions. We also study outer bi-Lipschitz extensions of near-isometric maps and show upper and lower bounds for them. Then, we present applications of our results to prioritized and terminal dimension reduction problems, described next. We prove a prioritized variant of the Johnson–Lindenstrauss lemma: given a set of points X ⊂ Rd of size N and a permutation (“priority ranking”) of X, there exists an embedding f of X into RO(log N) with distortion O(log log N) such that the point of rank j has only O(log3+ε j) non-zero coordinates – more specifically, all but the first O(log3+ε j) coordinates are equal to 0; the distortion of f restricted to the first j points (according to the ranking) is at most O(log log j). The result makes a progress towards answering an open question by Elkin, Filtser, and Neiman about prioritized dimension reductions. We prove that given a set X of N points in Rd, there exists a terminal dimension reduction embedding of Rd into Rd′, where d = O(log ε4 N), which preserves distances ∥x − ∥ between points x ∈ X and ∈ Rd, up to a multiplicative factor of 1 ± . This improves a recent result by Elkin, Filtser, and Neiman. The dimension reductions that we obtain are nonlinear, and this nonlinearity is necessary.

Original languageEnglish (US)
Title of host publicationSTOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
EditorsMonika Henzinger, David Kempe, Ilias Diakonikolas
PublisherAssociation for Computing Machinery
Pages574-586
Number of pages13
ISBN (Electronic)9781450355599
DOIs
StatePublished - Jun 20 2018
Event50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States
Duration: Jun 25 2018Jun 29 2018

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Other

Other50th Annual ACM Symposium on Theory of Computing, STOC 2018
CountryUnited States
CityLos Angeles
Period6/25/186/29/18

Keywords

  • Bi-lipschitz extension
  • Dimension reduction
  • Metric embedding
  • Near-isometric maps
  • Prioritized johnson-lindenstrauss

ASJC Scopus subject areas

  • Software

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