TY - GEN
T1 - Nonlinear dimension reduction via outer Bi-Lipschitz extensions
AU - Mahabadi, Sepideh
AU - Makarychev, Konstantin
AU - Makarychev, Yury
AU - Razenshteyn, Ilya
N1 - Funding Information:
Supported by NSF awards CAREER CCF-1150062, IIS-1302662, and CCF-1718820.
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
AN - SCOPUS:85049919796
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
T2 - 50th Annual ACM Symposium on Theory of Computing, STOC 2018
Y2 - 25 June 2018 through 29 June 2018
ER -