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

arXiv: https://arxiv.org/abs/2106.02154 · HTML: https://mlsanity.com/paper/2106.02154/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 114 references that are in the map.
- First posted on arXiv 3 Jun 2021.

## Builds on (most cited first, 16 of 16)

- [Graph Embedding Techniques, Applications, and Performance: A Survey](https://mlsanity.com/paper/1705.02801/) (2017)
- [DynGEM: Deep Embedding Method for Dynamic Graphs](https://mlsanity.com/paper/1805.11273/) (2018)
- [dyngraph2vec: Capturing Network Dynamics using Dynamic Graph Representation Learning](https://mlsanity.com/paper/1809.02657/) (2018)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Eigenvalue and Generalized Eigenvalue Problems: Tutorial](https://mlsanity.com/paper/1903.11240/) (2019)
- [Feature Selection and Feature Extraction in Pattern Analysis: A Literature Review](https://mlsanity.com/paper/1905.02845/) (2019)
- [Locally Linear Embedding and its Variants: Tutorial and Survey](https://mlsanity.com/paper/2011.10925/) (2020)
- [Multidimensional Scaling, Sammon Mapping, and Isomap: Tutorial and Survey](https://mlsanity.com/paper/2009.08136/) (2020)
- [Unsupervised and Supervised Principal Component Analysis: Tutorial](https://mlsanity.com/paper/1906.03148/) (2019)
- [Fisher and Kernel Fisher Discriminant Analysis: Tutorial](https://mlsanity.com/paper/1906.09436/) (2019)
- [Supervised Domain Adaptation: A Graph Embedding Perspective and a Rectified Experimental Protocol](https://mlsanity.com/paper/2004.11262/) (2020)
- [Supervised Domain Adaptation using Graph Embedding](https://mlsanity.com/paper/2003.04063/) (2020)
- [Locality Preserving Projections for Grassmann manifold](https://mlsanity.com/paper/1704.08458/) (2017)
- [On Eigenvalues of Laplacian Matrix for a Class of Directed Signed Graphs](https://mlsanity.com/paper/1705.04406/) (2017)
- [Laplacian Matrix for Dimensionality Reduction and Clustering](https://mlsanity.com/paper/1909.08381/) (2019)
- [Fast Spectral Clustering Using Autoencoders and Landmarks](https://mlsanity.com/paper/1704.02345/) (2017)

## Built on by (most cited first, 8 of 8)

- [Look Inside for More: Internal Spatial Modality Perception for 3D Anomaly Detection](https://mlsanity.com/paper/2412.13461/) (2024)
- [Uniform Manifold Approximation and Projection (UMAP) and its Variants: Tutorial and Survey](https://mlsanity.com/paper/2109.02508/) (2021)
- [Spectral, Probabilistic, and Deep Metric Learning: Tutorial and Survey](https://mlsanity.com/paper/2201.09267/) (2022)
- [Coupled Laplacian Eigenmaps for Locally-Aware 3D Rigid Point Cloud Matching](https://mlsanity.com/paper/2402.17372/) (2024)
- [A Tutorial on the Spectral Theory of Markov Chains](https://mlsanity.com/paper/2207.02296/) (2022)
- [Unified Framework for Spectral Dimensionality Reduction, Maximum Variance Unfolding, and Kernel Learning By Semidefinite Programming: Tutorial and Survey](https://mlsanity.com/paper/2106.15379/) (2021)
- [Sufficient Dimension Reduction for High-Dimensional Regression and Low-Dimensional Embedding: Tutorial and Survey](https://mlsanity.com/paper/2110.09620/) (2021)
- [Graph Dimensionality Reduction for Contextual Bandits: Structure-Specific Regret Bounds under Approximate Smoothness and Noisy Eigenspaces](https://mlsanity.com/paper/2606.27917/) (2026)

## 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.