Spectral embedding of weighted graphs

Gallagher, Ian; Jones, Andrew; Bertiger, Anna; Priebe, Carey E.; Rubin-Delanchy, Patrick

doi:10.48550/arxiv.1910.05534

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2021

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“…There are several situations, e.g. f any polynomial [41], the cosine kernel used in Section 4.1, the degree-corrected [26] or mixed-membership [6] stochastic block model, in which only the first p 0 (say) components of φ(•) are nonzero, where typically p 0 ≥ d. If, after n reaches p 0 , we embed into p = p 0 dimensions, we have [17] max i=1,...,n…”

Section: Introductionmentioning

confidence: 99%

Matrix factorisation and the interpretation of geodesic distance

Whiteley¹,

Gray²,

Rubin-Delanchy³

2021

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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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Section: Introductionmentioning

confidence: 99%