Metric Learning on Manifolds

TL;DR

This paper introduces a framework for distance metric learning on manifolds, enabling better representation of symbolic data like text and graphs, with improved clustering and classification results.

Contribution

It extends metric learning algorithms to a broad class of manifolds and derives sample complexity rates, addressing challenges of geometrical operations on curved spaces.

Findings

Improved $k$-means clustering on complex networks

Enhanced $k$-nearest neighbor classification accuracy

Framework applicable to a large class of manifolds

Abstract

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.