  We study the problem of supervised learning a metric space under
discriminative constraints. Given a universe $X$ and sets ${\cal S}, {\cal
D}\subset {X \choose 2}$ of similar and dissimilar pairs, we seek to find a
mapping $f:X\to Y$, into some target metric space $M=(Y,\rho)$, such that
similar objects are mapped to points at distance at most $u$, and dissimilar
objects are mapped to points at distance at least $\ell$. More generally, the
goal is to find a mapping of maximum accuracy (that is, fraction of correctly
classified pairs). We propose approximation algorithms for various versions of
this problem, for the cases of Euclidean and tree metric spaces. For both of
these target spaces, we obtain fully polynomial-time approximation schemes
(FPTAS) for the case of perfect information. In the presence of imperfect
information we present approximation algorithms that run in quasipolynomial
time (QPTAS). Our algorithms use a combination of tools from metric embeddings
and graph partitioning, that could be of independent interest.
