### 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 ⊂ R^{d} of size N and a permutation (“priority ranking”) of X, there exists an embedding f of X into R^{O}(log N) with distortion O(log log N) such that the point of rank j has only O(log^{3}+^{ε} j) non-zero coordinates – more specifically, all but the first O(log^{3}+^{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 R^{d}, there exists a terminal dimension reduction embedding of R^{d} into R^{d′}, where d^{′} = O(^{log}
_{ε4}
^{N}), which preserves distances ∥x − ∥ between points x ∈ X and ∈ R^{d}, 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 language | English (US) |
---|---|

Title of host publication | STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Monika Henzinger, David Kempe, Ilias Diakonikolas |

Publisher | Association for Computing Machinery |

Pages | 574-586 |

Number of pages | 13 |

ISBN (Electronic) | 9781450355599 |

DOIs | |

State | Published - Jun 20 2018 |

Event | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States Duration: Jun 25 2018 → Jun 29 2018 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
---|---|

ISSN (Print) | 0737-8017 |

### Other

Other | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 |
---|---|

Country | United States |

City | Los Angeles |

Period | 6/25/18 → 6/29/18 |

### Keywords

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

### ASJC Scopus subject areas

- Software

### Cite this

*STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing*(pp. 574-586). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/3188745.3188828

}

*STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing.*Proceedings of the Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, pp. 574-586, 50th Annual ACM Symposium on Theory of Computing, STOC 2018, Los Angeles, United States, 6/25/18. https://doi.org/10.1145/3188745.3188828

**Nonlinear dimension reduction via outer Bi-Lipschitz extensions.** / Mahabadi, Sepideh; Makarychev, Konstantin; Makarychev, Yury; Razenshteyn, Ilya.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Nonlinear dimension reduction via outer Bi-Lipschitz extensions

AU - Mahabadi, Sepideh

AU - Makarychev, Konstantin

AU - Makarychev, Yury

AU - Razenshteyn, Ilya

PY - 2018/6/20

Y1 - 2018/6/20

N2 - 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.

AB - 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.

KW - Bi-lipschitz extension

KW - Dimension reduction

KW - Metric embedding

KW - Near-isometric maps

KW - Prioritized johnson-lindenstrauss

UR - http://www.scopus.com/inward/record.url?scp=85049919796&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049919796&partnerID=8YFLogxK

U2 - 10.1145/3188745.3188828

DO - 10.1145/3188745.3188828

M3 - Conference contribution

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 574

EP - 586

BT - STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

A2 - Henzinger, Monika

A2 - Kempe, David

A2 - Diakonikolas, Ilias

PB - Association for Computing Machinery

ER -