  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.
