mlsanityData to 7 Oct 2026

Laplacian-Based Dimensionality Reduction Including Spectral Clustering, Laplacian Eigenmap, Locality Preserving Projection, Graph Embedding, and Diffusion Map: Tutorial and Survey

arXiv 2106.02154 ↗First posted 3 Jun 2021stat.ML cs.CV cs.LGcited by 8

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Abstract (arXiv)

This is a tutorial and survey paper for nonlinear dimensionality and feature extraction methods which are based on the Laplacian of graph of data. We first introduce adjacency matrix, definition of Laplacian matrix, and the interpretation of Laplacian. Then, we cover the cuts of graph and spectral clustering which applies clustering in a subspace of data. Different optimization variants of Laplacian eigenmap and its out-of-sample extension are explained. Thereafter, we introduce the locality preserving projection and its kernel variant as linear special cases of Laplacian eigenmap. Versions of graph embedding are then explained which are generalized versions of Laplacian eigenmap and locality preserving projection. Finally, diffusion map is introduced which is a method based on Laplacian of data and random walks on the data graph.

Key facts. Cited by 8 papers in the map (data through 7 Oct 2026). Builds on 16 of its 114 references that are in the map. First posted on arXiv 3 Jun 2021.

Builds on

16 of 114 references.

References · 16

  1. 2017Graph Embedding Techniques, Applications, and Performance: A Surveycited by 188
  2. 2018DynGEM: Deep Embedding Method for Dynamic Graphscited by 100
  3. 2018dyngraph2vec: Capturing Network Dynamics using Dynamic Graph Representation Learningcited by 88
  4. 2018Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operatorcited by 70
  5. 2019Eigenvalue and Generalized Eigenvalue Problems: Tutorialcited by 45
  6. 2019Feature Selection and Feature Extraction in Pattern Analysis: A Literature Reviewcited by 15
  7. 2020Locally Linear Embedding and its Variants: Tutorial and Surveycited by 14
  8. 2020Multidimensional Scaling, Sammon Mapping, and Isomap: Tutorial and Surveycited by 13
  9. 2019Unsupervised and Supervised Principal Component Analysis: Tutorialcited by 12
  10. 2019Fisher and Kernel Fisher Discriminant Analysis: Tutorialcited by 12
  11. 2020Supervised Domain Adaptation: A Graph Embedding Perspective and a Rectified Experimental Protocolcited by 8
  12. 2020Supervised Domain Adaptation using Graph Embeddingcited by 5
4 more
  1. 2017Locality Preserving Projections for Grassmann manifoldcited by 3
  2. 2017On Eigenvalues of Laplacian Matrix for a Class of Directed Signed Graphscited by 2
  3. 2019Laplacian Matrix for Dimensionality Reduction and Clusteringcited by 2
  4. 2017Fast Spectral Clustering Using Autoencoders and Landmarkscited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 3; 2022: 2; 2023: 0; 2024: 2; 2025: 0; 2026: 1.

8 papers.

  1. 2024Look Inside for More: Internal Spatial Modality Perception for 3D Anomaly Detectioncited by 30
  2. 2021Uniform Manifold Approximation and Projection (UMAP) and its Variants: Tutorial and Surveycited by 10
  3. 2022Spectral, Probabilistic, and Deep Metric Learning: Tutorial and Surveycited by 9
  4. 2024Coupled Laplacian Eigenmaps for Locally-Aware 3D Rigid Point Cloud Matchingcited by 3
  5. 2022A Tutorial on the Spectral Theory of Markov Chainscited by 2
  6. 2021Unified Framework for Spectral Dimensionality Reduction, Maximum Variance Unfolding, and Kernel Learning By Semidefinite Programming: Tutorial and Surveycited by 1
  7. 2021Sufficient Dimension Reduction for High-Dimensional Regression and Low-Dimensional Embedding: Tutorial and Surveycited by 1
  8. 2026Graph Dimensionality Reduction for Contextual Bandits: Structure-Specific Regret Bounds under Approximate Smoothness and Noisy Eigenspacescited by 0