Matrix factorisation and the interpretation of geodesic distance
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.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 DOI:
 10.48550/arXiv.2106.01260
 arXiv:
 arXiv:2106.01260
 Bibcode:
 2021arXiv210601260W
 Keywords:

 Statistics  Machine Learning;
 Computer Science  Machine Learning;
 62G05;
 62H20;
 62H12;
 62H30