  Are two sets of observations drawn from the same distribution? This problem
is a two-sample test. Kernel methods lead to many appealing properties. Indeed
state-of-the-art approaches use the $L^2$ distance between kernel-based
distribution representatives to derive their test statistics. Here, we show
that $L^p$ distances (with $p\geq 1$) between these distribution
representatives give metrics on the space of distributions that are
well-behaved to detect differences between distributions as they metrize the
weak convergence. Moreover, for analytic kernels, we show that the $L^1$
geometry gives improved testing power for scalable computational procedures.
Specifically, we derive a finite dimensional approximation of the metric given
as the $\ell_1$ norm of a vector which captures differences of expectations of
analytic functions evaluated at spatial locations or frequencies (i.e,
features). The features can be chosen to maximize the differences of the
distributions and give interpretable indications of how they differs. Using an
$\ell_1$ norm gives better detection because differences between
representatives are dense as we use analytic kernels (non-zero almost
everywhere). The tests are consistent, while much faster than state-of-the-art
quadratic-time kernel-based tests. Experiments on artificial and real-world
problems demonstrate improved power/time tradeoff than the state of the art,
based on $\ell_2$ norms, and in some cases, better outright power than even the
most expensive quadratic-time tests.
