Matrix factorisation and the interpretation of geodesic distance

Nick Whiteley, Annie Gray, Patrick Rubin-Delanchy

Keywords: graph, covariance matrix, nonlinear dimension reduction, manifold, embedding, matrix factorisation, Isomap

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.

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TL;DR: Obtain 'true' node representations using matrix factorisation followed by manifold learning

Code: https://github.com/anniegray52/graphs

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2106.01260/code)