Laplacian Matrix for Dimensionality Reduction and Clustering

TL;DR

This paper discusses how the Laplacian matrix of a graph can be used for dimensionality reduction and clustering in machine learning, providing foundational concepts and algorithms.

Contribution

It introduces the Laplacian matrix and presents algorithms leveraging it for data embedding and clustering.

Findings

Laplacian matrix effectively captures graph structure.

Algorithms based on Laplacian improve data embedding.

Laplacian-based methods enhance clustering performance.

Abstract

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.