  We consider the supervised classification problem of machine learning in
Cayley-Klein projective geometries: We show how to learn a curved Mahalanobis
metric distance corresponding to either the hyperbolic geometry or the elliptic
geometry using the Large Margin Nearest Neighbor (LMNN) framework. We report on
our experimental results, and further consider the case of learning a mixed
curved Mahalanobis distance. Besides, we show that the Cayley-Klein Voronoi
diagrams are affine, and can be built from an equivalent (clipped) power
diagrams, and that Cayley-Klein balls have Mahalanobis shapes with displaced
centers.
