  Data depth is a statistical function that generalizes order and quantiles to
the multivariate setting and beyond, with applications spanning over
descriptive and visual statistics, anomaly detection, testing, etc. The
celebrated halfspace depth exploits data geometry via an optimization program
to deliver properties of invariances, robustness, and non-parametricity.
Nevertheless, it implicitly assumes convex data supports and requires
exponential computational cost. To tackle distribution's multimodality, we
extend the halfspace depth in a Reproducing Kernel Hilbert Space (RKHS). We
show that the obtained depth is intuitive and establish its consistency with
provable concentration bounds that allow for homogeneity testing. The proposed
depth can be computed using manifold gradient making faster than halfspace
depth by several orders of magnitude. The performance of our depth is
demonstrated through numerical simulations as well as applications such as
anomaly detection on real data and homogeneity testing.
