Nonlinear Dimension Reduction via Outer Bi-Lipschitz Extensions

11/08/2018
by   Sepideh Mahabadi, et al.
0

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. * 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( N) with distortion O( N) such that the point of rank j has only O(^3 + ε j) non-zero coordinates - more specifically, all but the first O(^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( 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( 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.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset