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The performance of many machine learning techniques depends on the choice ofan appropriate similarity or distance measure on the input space. Similaritylearning (or metric learning) aims at building such a measure from trainingdata so that observations with the same (resp. different) label are as close(resp. far) as possible. In this paper, similarity learning is investigatedfrom the perspective of pairwise bipartite ranking, where the goal is to rankthe elements of a database by decreasing order of the probability that theyshare the same label with some query data point, based on the similarityscores. A natural performance criterion in this setting is pointwise ROCoptimization: maximize the true positive rate under a fixed false positiverate. We study this novel perspective on similarity learning through a rigorousprobabilistic framework. The empirical version of the problem gives rise to aconstrained optimization formulation involving U-statistics, for which wederive universal learning rates as well as faster rates under a noiseassumption on the data distribution. We also address the large-scale setting byanalyzing the effect of sampling-based approximations. Our theoretical resultsare supported by illustrative numerical experiments.

A Probabilistic Theory of Supervised Similarity Learning for Pointwise ROC Curve Optimization