# A tutorial on spectral clustering

@article{Luxburg2007ATO, title={A tutorial on spectral clustering}, author={Ulrike von Luxburg}, journal={Statistics and Computing}, year={2007}, volume={17}, pages={395-416} }

Abstract
In recent years, spectral clustering has become one of the most popular modern clustering algorithms. [... ] Key Method 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.
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## 8,294 Citations

The latest research progress on spectral clustering

- Computer ScienceNeural Computing and Applications
- 2013

The basic concepts of graph theory are introduced and main matrix representations of the graph are reviewed, then the objective functions of typical graph cut methods are compared, and the nature of spectral clustering algorithm is explored.

Research on Spectral Clustering

- Computer ScienceCIT 2014
- 2014

This paper describes different graph partition criteria, the definition of spectral clustering, and clustering steps, etc, and some improvements are introduced briefly in order to solve the disadvantage of spectral clusters.

A Note on Spectral Clustering Method Based on Normalized Cut Criterion

- Computer Science2009 Chinese Conference on Pattern Recognition
- 2009

A note is given on why the first k eigenvectors in the algorithm are chosen and the conditions for indicator vectors under which the clustering problem could lead to the problem of minimizing the objective function of the spectral clustering method based on normalized cut criterion.

An application of spectral clustering approach to detect communities in data modeled by graphs

- Computer ScienceNISS19
- 2019

An application of spectral clustering to detect communities in data from real world after modeling those data by graphs and a comparison between the obtained results using the unnormalized and the normalized algorithms are presented.

A randomized algorithm for spectral clustering

- Computer Science, MathematicsESANN
- 2010

A bound for choosing a correct number of eigenvectors in a randomized spectral algorithm able to find a clustering solution is shown and the efficacy of the algorithm is shown with experiments on real world graphs.

Spectral Clustering on a Budget

- Computer ScienceAISTATS
- 2011

This paper focuses on the problem of performing spectral clustering under a budget constraint, where there is a limit on the number of entries which can be queried from the similarity matrix.

Partitioning Well-Clustered Graphs: Spectral Clustering Works!

- Computer Science, MathematicsSIAM J. Comput.
- 2017

It is shown that spectral clustering gives a good approximation of the optimal clustering of graphs generated from stochastic models and a nearly linear time algorithm for partitioning well-clustered graphs based on computing a matrix exponential and approximate nearest neighbor data structures.

Spectral Clustering Survey HU , Pili

- Computer Science
- 2012

This article aims at providing systematic ways to explore new spectral clustering algorithms through the analysis of a bunch of algorithms, and gives several unifying views of Spectral Embedding Technique: graph framework, kernel framework, trace maximization.

Principal coordinate clustering

- Computer Science2017 IEEE International Conference on Big Data (Big Data)
- 2017

This paper introduces a clustering algorithm, called principal coordinate clustering. It takes in a similarity matrix SW of a data matrix W and computes the singular value decomposition of SW toâ€¦

A NEW SPECTRAL CLUSTERING APPROACH TO DETECTING COMMUNITIES IN GRAPHS

- Computer Science
- 2018

This paper presents an approach for the detection of communities from graphs with Spectral Clustering, and used igraph package in language R for simulation and implementation.

## References

SHOWING 1-10 OF 97 REFERENCES

Limits of Spectral Clustering

- Computer ScienceNIPS
- 2004

This paper investigates whether the partitions constructed on finite samples converge to a useful clustering of the whole data space as the sample size increases and concludes that while in the normalized case, spectral clustering usually converges to a nice partition of the data space, in the unnormalized case the same only holds under strong additional assumptions which are not always satisfied.

On Spectral Clustering: Analysis and an algorithm

- Computer ScienceNIPS
- 2001

A simple spectral clustering algorithm that can be implemented using a few lines of Matlab is presented, and tools from matrix perturbation theory are used to analyze the algorithm, and give conditions under which it can be expected to do well.

Consistency of spectral clustering

- Computer Science
- 2008

It is proved that one of the two major classes of spectral clustering (normalized clustering) converges under very general conditions, while the other is only consistent under strong additional assumptions, which are not always satisfied in real data.

Learning Spectral Clustering

- Computer ScienceNIPS
- 2003

A new cost function for spectral clustering is derived based on a measure of error between a given partition and a solution of the spectral relaxation of a minimum normalized cut problem.

On the Convergence of Spectral Clustering on Random Samples: The Normalized Case

- Computer Science, MathematicsCOLT
- 2004

It is shown that spectral clustering usually converges to an intuitively appealing limit partition of the data space and argues that in case of the unnormalized graph Laplacian, equally strong convergence results are difficult to obtain.

Kernel k-means: spectral clustering and normalized cuts

- Computer ScienceKDD
- 2004

The generality of the weighted kernel k-means objective function is shown, and the spectral clustering objective of normalized cut is derived as a special case, leading to a novel weightedkernel k-Means algorithm that monotonically decreases the normalized cut.

A Unified View of Kernel k-means , Spectral Clustering and Graph Cuts

- Computer Science
- 2004

Results show that non-spectral methods for graph partitioning are as effective as spectral methods and can be used for problems such as image segmentation in addition to data clustering.

A Sober Look at Clustering Stability

- Computer ScienceCOLT
- 2006

It is concluded that stability is not a well-suited tool to determine the number of clusters - it is determined by the symmetries of the data which may be unrelated to clustering parameters.

On the Quality of Spectral Separators

- Mathematics, Computer Science
- 1998

This work considers two popular spectral separator algorithms and provides counterexamples showing that these algorithms perform poorly on certain graphs, and introduces some facts about the structure of eigenvectors of certain types of Laplacian and symmetric matrices.

A Random Walks View of Spectral Segmentation

- Computer ScienceAISTATS
- 2001

It is proved that the Normalized Cut method arises naturally from the framework and a complete characterization of the cases when the Normalization Cut algorithm is exact is provided.