A Tutorial on Spectral Clustering (0711.0189v1)

Abstract: In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.

• The paper demonstrates how spectral clustering leverages graph Laplacians to relax NP-hard graph partitioning problems into eigenvalue computations.
• It introduces several spectral clustering methods, detailing both unnormalized and normalized approaches alongside the practical use of the eigengap heuristic.
• The tutorial emphasizes robust implementation practices, including efficient eigenvector computation and careful similarity graph construction, to enhance clustering performance.

Spectral Clustering: An In-depth Tutorial

Introduction

Spectral clustering has emerged as a significant method in modern clustering algorithms due to its simplicity, efficiency, and often superior performance compared to traditional techniques like k-means. The paper "A Tutorial on Spectral Clustering" by Ulrike von Luxburg provides a detailed and structured introduction to spectral clustering algorithms, discussing their mathematical foundations, practical implementations, and conceptual explanations.

Mathematical Foundations

The essence of spectral clustering is grounded in graph theory and linear algebra. It operates by constructing a similarity graph from a dataset and then partitioning this graph. The fundamental tools employed are the graph Laplacians, which come in different flavors: the unnormalized graph Laplacian, denoted as $L = D - W$, and the normalized graph Laplacians, denoted as $L_{sym} = D^{-1/2} L D^{-1/2}$ and $L_{rw} = D^{-1} L$. Here, $D$ is the degree matrix, and $W$ is the weighted adjacency matrix of the graph.

Spectral Clustering Algorithms

Three primary spectral clustering algorithms are discussed in the tutorial, each utilizing a different form of the graph Laplacian:

1. Unnormalized Spectral Clustering: Directly uses the eigenvectors of the unnormalized Laplacian $L$.
2. Normalized Spectral Clustering according to Shi and Malik (2000): Employs the first $k$ generalized eigenvectors of $Lx = \lambda D x$, effectively working with $L_{rw}$.
3. Normalized Spectral Clustering according to Ng, Jordan, and Weiss (2002): Operates on the normalized Laplacian $L_{sym}$, augmented with a row normalization step in the final embedding space.

Graph Partitioning Perspective

Spectral clustering can be interpreted as an approximation to graph partitioning problems such as RatioCut and normalized cut (Ncut). By relaxing these NP-hard partitioning problems into continuous optimization problems, spectral clustering transforms the discrete clustering task into a well-defined eigenvalue problem. For instance, minimizing RatioCut can be approximated by using the eigenvectors corresponding to the smallest eigenvalues of $L$.

Random Walks and Perturbation Theory

The paper presents a compelling interpretation of spectral clustering through random walks on the similarity graph. By minimizing Ncut, spectral clustering ensures that a random walk stays within the same cluster and seldom transitions between clusters. The relationship between spectral clustering and the commute distance on the graph further elucidates its effectiveness in separating densely connected components.

Perturbation theory offers another lens to understand spectral clustering, particularly in nearly ideal cases where clusters are almost disconnected. The Davis-Kahan theorem is utilized to demonstrate that the eigenvectors of a perturbed Laplacian matrix remain close to those of the ideal unperturbed case. This implies that spectral clustering can robustly detect clusters even under slight perturbations in the data.

Practical Considerations

The tutorial emphasizes the importance of constructing an appropriate similarity graph. Choices include k-nearest neighbor graphs, epsilon-neighborhood graphs, mutual k-nearest neighbor graphs, and fully connected graphs with Gaussian similarity, each with its own advantages and parameter selection challenges. The stability and performance of spectral clustering are highly sensitive to these choices.

Additionally, computing the eigenvectors of the Laplacian matrix efficiently, especially for large and sparse graphs, is crucial. Techniques like the Lanczos method ensure computational feasibility.

Choosing the Number of Clusters

The eigengap heuristic is recommended for determining the optimal number of clusters $k$. By examining the eigenvalues of the graph Laplacian, one can identify a significant gap between the eigenvalues, suggesting a natural partitioning of the graph into $k$ clusters.

Which Laplacian to Use?

The paper advocates for the use of normalized spectral clustering with $L_{sym}$ due to its balanced objectives and consistency properties. While both normalized Laplacians ($L_{sym}$ and $L_{rw}$) address fundamental clustering goals, $L_{sym}$'s eigenvectors are direct cluster indicators, offering a clearer and more interpretable embedding space.

Conclusion and Future Directions

The tutorial by Ulrike von Luxburg offers a comprehensive guide to understanding and implementing spectral clustering. It bridges theoretical insights with practical algorithms, providing a robust framework for tackling various clustering problems. As the field evolves, further research on the interplay between graph construction parameters and clustering performance, along with advancements in eigenvector computation techniques, will continue to enhance the utility of spectral clustering in diverse applications.