Matrix factorisation and the interpretation of geodesic distance

• URL: http://arxiv.org/abs/2106.01260v1
• Date: Wed, 2 Jun 2021 16:11:33 GMT
• Title: Matrix factorisation and the interpretation of geodesic distance
• Authors: Nick Whiteley, Annie Gray and Patrick Rubin-Delanchy
• Abstract summary: Given a graph or similarity matrix, we consider the problem of recovering a notion of true distance between the nodes. We show that this can be accomplished in two steps: matrix factorisation, followed by nonlinear dimension reduction.
• Score: 6.445605125467574
• License: http://creativecommons.org/licenses/by/4.0/
• 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. Through new insights into the manifold geometry underlying a generic latent position model, 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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