# Laplacian Matrix for Dimensionality Reduction and Clustering

arXiv: https://arxiv.org/abs/1909.08381 · HTML: https://mlsanity.com/paper/1909.08381/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Sep 2019.

## Built on by (most cited first, 2 of 2)

- [Laplacian-Based Dimensionality Reduction Including Spectral Clustering, Laplacian Eigenmap, Locality Preserving Projection, Graph Embedding, and Diffusion Map: Tutorial and Survey](https://mlsanity.com/paper/2106.02154/) (2021)
- [A Tutorial on the Spectral Theory of Markov Chains](https://mlsanity.com/paper/2207.02296/) (2022)

## Abstract (arXiv)

Many problems in machine learning can be expressed by means of a graph with nodes representing training samples and edges representing the relationship between samples in terms of similarity, temporal proximity, or label information. Graphs can in turn be represented by matrices. A special example is the Laplacian matrix, which allows us to assign each node a value that varies only little between strongly connected nodes and more between distant nodes. Such an assignment can be used to extract a useful feature representation, find a good embedding of data in a low dimensional space, or perform clustering on the original samples. In these lecture notes we first introduce the Laplacian matrix and then present a small number of algorithms designed around it.