A Bipartite Ranking Approach to the Two-Sample Problem

The two-sample problem, which consists in testing whether independent samples on $\mathbb{R}^d$ are drawn from the same (unknown) distribution, finds applications in many areas. Its study in high-dimension is the subject of much attention, especially because the information acquisition processes at work in the Big Data era often involve various sources, poorly controlled, leading to datasets possibly exhibiting a strong sampling bias. While classic methods relying on the computation of a discrepancy measure between the empirical distributions face the curse of dimensionality, we develop an alternative approach based on statistical learning and extending rank tests, capable of detecting small departures from the null assumption in the univariate case when appropriately designed. Overcoming the lack of natural order on $\mathbb{R}^d$ when $d\geq 2$, it is implemented in two steps. Assigning to each of the samples a label (positive vs. negative) and dividing them into two parts, a preorder on $\mathbb{R}^d$ defined by a real-valued scoring function is learned by means of a bipartite ranking algorithm applied to the first part and a rank test is applied next to the scores of the remaining observations to detect possible differences in distribution. Because it learns how to project the data onto the real line nearly like (any monotone transform of) the likelihood ratio between the original multivariate distributions would do, the approach is not much affected by the dimensionality, ignoring ranking model bias issues, and preserves the advantages of univariate rank tests. Nonasymptotic error bounds are proved based on recent concentration results for two-sample linear rank-processes and an experimental study shows that the approach promoted surpasses alternative methods standing as natural competitors.