Approximating Lp unit balls via random sampling

Abstract

Let X be an isotropic random vector in Rd that satisfies that for every v∈Sd−1, ‖〈X,v〉‖Lq≤L‖〈X,v〉‖Lp for some q≥2p. We show that for 0<ε<1, a set of N=c(p,q,ε)d random points, selected independently according to X, can be used to construct a 1±ε approximation of the Lp unit ball endowed on Rd by X. Moreover, c(p,q,ε)≤cpε−2log⁡(2/ε); when q=2p the approximation is achieved with probability at least 1−2exp⁡(−cNε2/log2⁡(2/ε)) and if q is much larger than p—say, q=4p, the approximation is achieved with probability at least 1−2exp⁡(−cNε2).In particular, when X is a log-concave random vector, this estimate improves the previous state-of-the-art—that N=c′(p,ε)dp/2log⁡d random points are enough and that the approximation is valid with constant probability.