Weak - nets have basis of size O ( 1 / log ( 1 / ) ) in any dimension

Given a set P of n points in Rd and > 0, we consider the problem of constructing weak -nets for P . We show the following: pick a random sample Q of size O(1/ log (1/ )) from P . Then, with constant probability, a weak -net of P can be constructed from only the points of Q. This shows that weak -nets in Rd can be computed from a subset of P of size O(1/ log(1/ )) with only the constant of proportionality depending on the dimension, unlike all previous work where the size of the subset had the dimension in the exponent of 1/ . However, our final weak -nets still have a large size (with the dimension appearing in the exponent of 1/ ). © 2007 Elsevier B.V. All rights reserved.