Large Margin Nearest Neighbor Classification using Curved Mahalanobis Distances

TL;DR

This paper introduces a method to learn curved Mahalanobis distances in Cayley-Klein geometries for improved nearest neighbor classification, demonstrating affine Voronoi diagrams and Mahalanobis-shaped balls.

Contribution

It extends LMNN to curved geometries, enabling learning of hyperbolic and elliptic metrics, and analyzes geometric properties of Cayley-Klein Voronoi diagrams and balls.

Findings

Successful learning of hyperbolic and elliptic Mahalanobis distances

Cayley-Klein Voronoi diagrams are affine and derived from power diagrams

Cayley-Klein balls have Mahalanobis shapes with displaced centers

Abstract

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.