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 e-prints
- Pub Date:
- June 2021
- DOI:
- 10.48550/arXiv.2106.01260
- arXiv:
- arXiv:2106.01260
- Bibcode:
- 2021arXiv210601260W
- Keywords:
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- Statistics - Machine Learning;
- Computer Science - Machine Learning;
- 62G05;
- 62H20;
- 62H12;
- 62H30