Nonlinear dimension reduction via outer Bi-Lipschitz extensions

Sepideh Mahabadi, Konstantin Makarychev, Yury Makarychev, Ilya P. Razenshteyn

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 (loglog 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 (loglog 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 N /ε 4 ), which preserves distances || x − y || between points x ∈ X and y ∈ ℝ d , up to a multiplicative factor of 1 ± ε. This improves a recent result by Elkin, Filtser, and Neiman. The dimension reductions that we obtain are nonlinear, and this nonlinearity is necessary.