A Bipartite Ranking Approach to the Two-Sample Problem

The two-sample problem consists in testing whether two independent samples are drawn from the same (unknown) probability distribution. It finds applications in many areas, ranging from clinical trials to data attribute matching. Its study in high-dimension is the subject of much attention, in particular as the information acquisition processes can involve various sources being often poorly controlled, possibly leading to datasets with strong sampling bias that may jeopardize their statistical analysis. While classic methods relying on a discrepancy measure between empirical versions of the distributions face the curse of dimensionality, we develop an alternative approach based on statistical learning and extending rank tests, known to be asymptotically optimal for univariate data when appropriately designed. Overcoming the lack of natural order on high-dimension, it is implemented in two steps. Assigning a label to each sample, and dividing them into two halves, a preorder on the feature space defined by a real-valued scoring function is learned by a bipartite ranking algorithm applied to the first halves. Next, a two-sample homogeneity rank test is applied to the (univariate) scores of the remaining observations. Because it learns how to map the data onto the real line like (any monotone transform of) the likelihood ratio between the original multivariate distributions, the approach is not affected by the dimensionality, ignores ranking model bias issues, and preserves the asymptotic optimality of univariate R-tests, capable of detecting small departures from the null assumption. Beyond a theoretical analysis establishing nonasymptotic bounds for the two types of error of the method based on recent concentration results for two-sample linear R-processes, an extensive experimental study shows higher performance of the proposed method compared to classic ones.