  The problem of robust hypothesis testing is studied, where under the null and
the alternative hypotheses, the data-generating distributions are assumed to be
in some uncertainty sets, and the goal is to design a test that performs well
under the worst-case distributions over the uncertainty sets. In this paper,
uncertainty sets are constructed in a data-driven manner using kernel method,
i.e., they are centered around empirical distributions of training samples from
the null and alternative hypotheses, respectively; and are constrained via the
distance between kernel mean embeddings of distributions in the reproducing
kernel Hilbert space, i.e., maximum mean discrepancy (MMD). The Bayesian
setting and the Neyman-Pearson setting are investigated. For the Bayesian
setting where the goal is to minimize the worst-case error probability, an
optimal test is firstly obtained when the alphabet is finite. When the alphabet
is infinite, a tractable approximation is proposed to quantify the worst-case
average error probability, and a kernel smoothing method is further applied to
design test that generalizes to unseen samples. A direct robust kernel test is
also proposed and proved to be exponentially consistent. For the Neyman-Pearson
setting, where the goal is to minimize the worst-case probability of miss
detection subject to a constraint on the worst-case probability of false alarm,
an efficient robust kernel test is proposed and is shown to be asymptotically
optimal. Numerical results are provided to demonstrate the performance of the
proposed robust tests.
