RISE: Rank in Similarity Graph Edge-Count Two-Sample Test

Two-sample hypothesis testing for high-dimensional data is ubiquitous nowadays. Rank-based tests are popular nonparametric methods for univariate data. However, they are difficult to be extended to high-dimensional data. In this paper, we propose a new non-parametric two-sample testing procedure, Rank In Similarity graph Edge-count two-sample test (RISE). The new test statistic is constructed on a rank-weighted similarity graph, such as the $k$-nearest neighbor graph. As a result, RISE can also be applied to non-Euclidean data. Theoretically, we prove that, under some mild conditions, the new test statistic converges to the $\chi_2^2$ distribution under the permutation null distribution, enabling a fast type-I error control. RISE exhibits good power under a wide range of alternatives compared to existing methods, as shown in extensive simulations. The new test is illustrated on the New York City taxi data for comparing travel patterns in consecutive months and a brain network dataset in comparing male and female subjects.