Matrix factorisation and the interpretation of geodesic distance
Machine Learning
2022-09-23 v3 Machine Learning
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.
Keywords
Cite
@article{arxiv.2106.01260,
title = {Matrix factorisation and the interpretation of geodesic distance},
author = {Nick Whiteley and Annie Gray and Patrick Rubin-Delanchy},
journal= {arXiv preprint arXiv:2106.01260},
year = {2022}
}