Matrix factorisation and the interpretation of geodesic distance

TL;DR

This paper presents a method combining matrix factorisation and nonlinear dimension reduction to recover true latent distances and positions in graph or similarity data, effectively approximating geodesic distances.

Contribution

It demonstrates that a two-step process of matrix factorisation and nonlinear embedding can accurately recover latent positions from similarity matrices.

Findings

Spectral embedding followed by Isomap effectively recovers latent positions.

The method approximates geodesic distances with encouraging experimental results.

Combining matrix factorisation with nonlinear reduction is a promising approach.

Abstract

Given a graph or similarity matrix, we consider the problem of recovering a notion of true distance between the nodes, and so their true positions. We show that this can be accomplished in two steps: matrix factorisation, followed by nonlinear dimension reduction. This combination is effective because the point cloud obtained in the first step lives close to a manifold in which latent distance is encoded as geodesic distance. Hence, a nonlinear dimension reduction tool, approximating geodesic distance, can recover the latent positions, up to a simple transformation. We give a detailed account of the case where spectral embedding is used, followed by Isomap, and provide encouraging experimental evidence for other combinations of techniques.