linear algebra - Sparse approximate representation of a collection of vectors

By triangle inequality, preserving the property you wish for means that you can find "representatives" for each $v$ so that the $ell_1$ distances between any $v, v'$ are preserved to within 2$epsilon$ additive error.

There is a general result by Brinkman and Charikar that says that in general, for a collection of $n$ vectors in an $ell_1$ space, there's no way to construct a set of $n$ vectors in a smaller (e.g $log n$) dimensional space) that preserves distances approximately even multiplicatively (let alone additively). This distinction is important if you have vectors in the original space that are $O(epsilon)$ apart, but otherwise doesn't matter greatly.

Brinkman, B. and Charikar, M. 2005. On the impossibility of dimension reduction in l1. J. ACM 52, 5 (Sep. 2005), 766-788. DOI= http://doi.acm.org/10.1145/1089023.1089026

So I'm guessing that the answer to your first question should be no.