  The performance of many machine learning techniques depends on the choice of
an appropriate similarity or distance measure on the input space. Similarity
learning (or metric learning) aims at building such a measure from training
data so that observations with the same (resp. different) label are as close
(resp. far) as possible. In this paper, similarity learning is investigated
from the perspective of pairwise bipartite ranking, where the goal is to rank
the elements of a database by decreasing order of the probability that they
share the same label with some query data point, based on the similarity
scores. A natural performance criterion in this setting is pointwise ROC
optimization: maximize the true positive rate under a fixed false positive
rate. We study this novel perspective on similarity learning through a rigorous
probabilistic framework. The empirical version of the problem gives rise to a
constrained optimization formulation involving U-statistics, for which we
derive universal learning rates as well as faster rates under a noise
assumption on the data distribution. We also address the large-scale setting by
analyzing the effect of sampling-based approximations. Our theoretical results
are supported by illustrative numerical experiments.
