  We develop a kernel projected Wasserstein distance for the two-sample test,
an essential building block in statistics and machine learning: given two sets
of samples, to determine whether they are from the same distribution. This
method operates by finding the nonlinear mapping in the data space which
maximizes the distance between projected distributions. In contrast to existing
works about projected Wasserstein distance, the proposed method circumvents the
curse of dimensionality more efficiently. We present practical algorithms for
computing this distance function together with the non-asymptotic uncertainty
quantification of empirical estimates. Numerical examples validate our
theoretical results and demonstrate good performance of the proposed method.
