mlsanityData to 7 Oct 2026

Matrix factorisation and the interpretation of geodesic distance

arXiv 2106.01260 ↗First posted 2 Jun 2021stat.ML cs.LGcited by 6

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Abstract (arXiv)

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.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 22 of its 82 references that are in the map. First posted on arXiv 2 Jun 2021.

Builds on

22 of 82 references.

References · 22

  1. 2018UMAP: Uniform Manifold Approximation and Projection for Dimension Reductioncited by 2,841
  2. 2016node2vec: Scalable Feature Learning for Networkscited by 1,913
  3. 2017Community Detection and Stochastic Block Modelscited by 317
  4. 2017Network Embedding as Matrix Factorization: Unifying DeepWalk, LINE, PTE, and node2veccited by 197
  5. 2013Consistency of spectral clustering in stochastic block modelscited by 128
  6. 2017Statistical inference on random dot product graphs: a surveycited by 70
  7. 2017A statistical interpretation of spectral embedding: the generalised random dot product graphcited by 55
  8. 2017Rates of Convergence of Spectral Methods for Graphon Estimationcited by 28
  9. 2018On a 'Two Truths' Phenomenon in Spectral Graph Clusteringcited by 24
  10. 2013Perfect Clustering for Stochastic Blockmodel Graphs via Adjacency Spectral Embeddingcited by 23
  11. 2015Community Detection and Classification in Hierarchical Stochastic Blockmodelscited by 23
  12. 2020The multilayer random dot product graphcited by 18
10 more
  1. 2016Limit theorems for eigenvectors of the normalized Laplacian for random graphscited by 17
  2. 2020Manifold structure in graph embeddingscited by 17
  3. 2018Network Representation Using Graph Root Distributionscited by 12
  4. 2019Latent Distance Estimation for Random Geometric Graphscited by 11
  5. 2018On estimation and inference in latent structure random graphscited by 9
  6. 2019Spectral embedding of weighted graphscited by 9
  7. 2020Spectral clustering on spherical coordinates under the degree-corrected stochastic blockmodelcited by 9
  8. 2020Learning 1-Dimensional Submanifolds for Subsequent Inference on Random Dot Product Graphscited by 6
  9. 2021Spectral clustering under degree heterogeneity: a case for the random walk Laplaciancited by 6
  10. 2019Comparison of brain connectomes using geodesic distance on manifold:a twin studycited by 2

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 2; 2023: 0; 2024: 0; 2025: 1; 2026: 3.

6 papers.

  1. 2022Implications of sparsity and high triangle density for graph representation learningcited by 1
  2. 2022Spectral embedding and the latent geometry of multipartite networkscited by 0
  3. 2025How high is `high'? Rethinking the roles of dimensionality in topological data analysis and manifold learningcited by 0
  4. 2026Uncovering Locally Low-dimensional Structure in Networks by Locally Optimal Spectral Embeddingcited by 0
  5. 2026SalAngaBhava: A Sinhala Market Dataset for Aspect-based Sentiment Analysiscited by 0
  6. 2026Learning Submanifolds for Subsequent Inference on Random Dot Product Graphs, Part 1: Theorycited by 0