Linear Dimension Reduction Approximately Preserving Level-Sets of the 1-Norm

Abstract

We choose a family of matrices F : \R^D \to \R^k and a metric \rho on \R^k such that with highprobability, \rho(F (x), F (y)) is a strictly concave increasing function of ||x − y||_1 > 8 \epsilon^2for x, y \in \R^D , up to a multiplicative error of 1 ±\epsilon. In particular, if X is a set of Npoints in \R^D , the target dimension k may be chosen as C ln^2 (N^{c+2})/(\epsilon^2(1 −\epsilon )^2), withC a constant and \epsilon > N^{−c} , to ensure all pairs of points of X of distance at least 8\epsilon^2are treated this way, with failure probability at most N^{-c} for c > 1. In some cases,distances smaller than 8\epsilon^2 can also be addressed. For distances larger than \sqrt{1 +\epsilon} ,the target dimension can be reduced to C ln(N^{c+2})/(\epsilon^2(1 −\epsilon )^2).