  Given a set of dissimilarity measurements amongst data points, determining
what metric representation is most "consistent" with the input measurements or
the metric that best captures the relevant geometric features of the data is a
key step in many machine learning algorithms. Existing methods are restricted
to specific kinds of metrics or small problem sizes because of the large number
of metric constraints in such problems. In this paper, we provide an active set
algorithm, Project and Forget, that uses Bregman projections, to solve metric
constrained problems with many (possibly exponentially) inequality constraints.
We provide a theoretical analysis of \textsc{Project and Forget} and prove that
our algorithm converges to the global optimal solution and that the $L_2$
distance of the current iterate to the optimal solution decays asymptotically
at an exponential rate. We demonstrate that using our method we can solve large
problem instances of three types of metric constrained problems: general weight
correlation clustering, metric nearness, and metric learning; in each case,
out-performing the state of the art methods with respect to CPU times and
problem sizes.
