  Two-sample hypothesis testing is a fundamental problem with various
applications, which faces new challenges in the high-dimensional context. To
mitigate the issue of the curse of dimensionality, high-dimensional data are
typically assumed to lie on a low-dimensional manifold. To incorporate
geometric informtion in the data, we propose to apply the Delaunay
triangulation and develop the Delaunay weight to measure the geometric
proximity among data points. In contrast to existing similarity measures that
only utilize pairwise distances, the Delaunay weight can take both the distance
and direction information into account. A detailed computation procedure to
approximate the Delaunay weight for the unknown manifold is developed. We
further propose a novel nonparametric test statistic using the Delaunay weight
matrix to test whether the underlying distributions of two samples are the same
or not. Applied on simulated data, the new test exhibits substantial power gain
in detecting differences in principal directions between distributions. The
proposed test also shows great power on a real dataset of human face images.
