Kernel Two-Sample Hypothesis Testing Using Kernel Set Classification

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.