  A method for extracting multiscale geometric features from a data cloud is
proposed and analyzed. The basic idea is to map each pair of data points into a
real-valued feature function defined on $[0,1]$. The construction of these
feature functions is heavily based on geometric considerations, which has the
benefits of enhancing interpretability. Further statistical analysis is then
based on the collection of the feature functions. The potential of the method
is illustrated by different applications, including classification of
high-dimensional and non-Euclidean data. For continuous data in Euclidean
space, our feature functions contain information about the underlying density
at a given base point (small scale features), and also about the depth of the
base point (large scale feature). As shown by our theoretical investigations,
the method combats the curse of dimensionality, and also shows some
adaptiveness towards sparsity. Connections to other concepts, such as random
set theory, localized depth measures and nonlinear multidimensional scaling,
are also explored.
