  The Maximum Mean Discrepancy (MMD) has been the state-of-the-art
nonparametric test for tackling the two-sample problem. Its statistic is given
by the difference in expectations of the witness function, a real-valued
function defined as a weighted sum of kernel evaluations on a set of basis
points. Typically the kernel is optimized on a training set, and hypothesis
testing is performed on a separate test set to avoid overfitting (i.e., control
type-I error). That is, the test set is used to simultaneously estimate the
expectations and define the basis points, while the training set only serves to
select the kernel and is discarded. In this work, we propose to use the
training data to also define the weights and the basis points for better data
efficiency. We show that 1) the new test is consistent and has a
well-controlled type-I error; 2) the optimal witness function is given by a
precision-weighted mean in the reproducing kernel Hilbert space associated with
the kernel; and 3) the test power of the proposed test is comparable or exceeds
that of the MMD and other modern tests, as verified empirically on challenging
synthetic and real problems (e.g., Higgs data).
