  The two-sample hypothesis testing problem is studied for the challenging
scenario of high dimensional data sets with small sample sizes. We show that
the two-sample hypothesis testing problem can be posed as a one-class set
classification problem. In the set classification problem the goal is to
classify a set of data points that are assumed to have a common class. We prove
that the average probability of error given a set is less than or equal to the
Bayes error and decreases as a power of $n$ number of sample data points in the
set. We use the positive definite Set Kernel for directly mapping sets of data
to an associated Reproducing Kernel Hilbert Space, without the need to learn a
probability distribution. We specifically solve the two-sample hypothesis
testing problem using a one-class SVM in conjunction with the proposed Set
Kernel. We compare the proposed method with the Maximum Mean Discrepancy,
F-Test and T-Test methods on a number of challenging simulated high dimensional
and small sample size data. We also perform two-sample hypothesis testing
experiments on six cancer gene expression data sets and achieve zero type-I and
type-II error results on all data sets.
