Matrix factorisation and the interpretation of geodesic distance
Nick Whiteley, Annie Gray, Patrick Rubin-Delanchy
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.
BibTeX
@inproceedings{
whiteley2021matrix,
title={Matrix factorisation and the interpretation of geodesic distance},
author={Nick Whiteley and Annie Gray and Patrick Rubin-Delanchy},
booktitle={Advances in Neural Information Processing Systems},
editor={A. Beygelzimer and Y. Dauphin and P. Liang and J. Wortman Vaughan},
year={2021},
url={https://openreview.net/forum?id=gwP8pc1OgN_}
}