Robust Hypothesis Testing with Kernel Uncertainty Sets

In this paper, the robust hypothesis testing problem is investigated, where under the null and the alternative hypotheses, the distributions are assumed to be in some uncertainty sets. The uncertainty sets are constructed in a data-driven manner, i.e., they are centered around empirical distributions. The distance between kernel mean embeddings of distributions in the reproducing kernel Hilbert space is used as the distance metric of uncertainty sets. The Bayesian setting is studied, where the goal is to minimize the worst-case error probability. An optimal test is firstly obtained for the case with a finite alphabet. For the case with an infinite alphabet, a tractable approximation is proposed to quantify the worst-case error probability, and a kernel smoothing method is further applied to design test that generalizes to unseen samples. A heuristic robust kernel test is also proposed and proved to be exponentially consistent. Numerical results are provided to demonstrate the performance of the proposed tests.