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

TL;DR

This paper provides a comprehensive tutorial and survey of Laplacian-based nonlinear dimensionality reduction methods, including spectral clustering, Laplacian eigenmap, and diffusion maps, explaining their theory, variants, and applications.

Contribution

It offers an in-depth overview of Laplacian-based methods, detailing their mathematical foundations, variants, and extensions, serving as a valuable resource for researchers and practitioners.

Findings

Detailed explanation of Laplacian eigenmap and its variants

Comparison of linear and nonlinear Laplacian methods

Introduction to diffusion maps and graph embedding techniques

Abstract

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.