Large Margin Nearest Neighbor Classification using Curved Mahalanobis Distances
This work addresses classification challenges in non-Euclidean geometries, offering a novel metric learning approach that is incremental to existing LMNN methods.
The paper tackles the supervised classification problem by learning curved Mahalanobis distances in hyperbolic and elliptic geometries using the Large Margin Nearest Neighbor framework, reporting experimental results and extending to mixed distances. It also demonstrates that Cayley-Klein Voronoi diagrams are affine and derivable from power diagrams, with balls having Mahalanobis shapes and displaced centers.
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.