  Recent literature has shown that symbolic data, such as text and graphs, is
often better represented by points on a curved manifold, rather than in
Euclidean space. However, geometrical operations on manifolds are generally
more complicated than in Euclidean space, and thus many techniques for
processing and analysis taken for granted in Euclidean space are difficult on
manifolds. A priori, it is not obvious how we may generalize such methods to
manifolds. We consider specifically the problem of distance metric learning,
and present a framework that solves it on a large class of manifolds, such that
similar data are located in closer proximity with respect to the manifold
distance function. In particular, we extend the existing metric learning
algorithms, and derive the corresponding sample complexity rates for the case
of manifolds. Additionally, we demonstrate an improvement of performance in
$k$-means clustering and $k$-nearest neighbor classification on real-world
complex networks using our methods.
