Nonlinear Metric Learning through Geodesic Interpolation within Lie Groups

TL;DR

This paper introduces a nonlinear metric learning method that uses geodesic interpolation on Lie groups to create smooth, diffeomorphic distance metrics, improving k-NN classification performance.

Contribution

The paper presents a novel nonlinear metric learning approach based on geodesic interpolation within Lie groups, ensuring smooth, diffeomorphic transformations for better distance metrics.

Findings

Effective on synthetic datasets

Improves k-NN classification accuracy

Demonstrates smooth, spatially varying metrics

Abstract

In this paper, we propose a nonlinear distance metric learning scheme based on the fusion of component linear metrics. Instead of merging displacements at each data point, our model calculates the velocities induced by the component transformations, via a geodesic interpolation on a Lie transfor- mation group. Such velocities are later summed up to produce a global transformation that is guaranteed to be diffeomorphic. Consequently, pair-wise distances computed this way conform to a smooth and spatially varying metric, which can greatly benefit k-NN classification. Experiments on synthetic and real datasets demonstrate the effectiveness of our model.