  Starting with Tukey's pioneering work in the 1970's, the notion of depth in
statistics has been widely extended especially in the last decade. These
extensions include high dimensional data, functional data, and manifold-valued
data. In particular, in the learning paradigm, the depth-depth method has
become a useful technique. In this paper we extend the notion of lens depth to
the case of data in metric spaces, and prove its main properties, with
particular emphasis on the case of Riemannian manifolds, where we extend the
concept of lens depth in such a way that it takes into account non-convex
structures on the data distribution. Next we illustrate our results with some
simulation results and also in some interesting real datasets, including
pattern recognition in phylogenetic trees using the depth--depth approach.
