Spectral Clustering

指導教授 : 王聖智 S. J. Wang學生 : 羅介暐 Jie-Wei Luo

Outline• Motivation

• Graph overview

• Spectral Clustering

• Another point of view

• Conclusion

Motivation

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Þ K-means performs very poorly in this space due Dataset exhibits complex cluster shapes

K-means

Spectral Clustering

Scatter plot of a 2D data set

K-means Clustering Spectral Clustering

U. von Luxburg. A tutorial on spectral clustering. Technical report, Max Planck Institute for Biological Cybernetics, Germany, 2007.

Graph overview

Graph Partitioning

Graph notation

Graph Cut

Distance and Similarity

Graph partitioning

First-graph representation of data

Then-graph partitioning

In this talk–mainly how to find a good partitioning of a given graph using spectral properties of that graph

Graph notation

Note: 1. Sij0 2. Sij=Sji

Always assume that similarities sij are symmetric, non-negativeThen graph is undirected, can be weighted

Graph notation

Degree of vertex vi є V

𝑑𝑖=∑𝑗=1

𝑛

𝑤𝑖𝑗 vi

𝑣𝑜𝑙 ( 𝐴 )=∑𝑖𝜖 𝐴

𝑑𝑖

𝑐𝑢𝑡 ( 𝐴 ,𝐵 )= ∑𝑖 𝜖 𝐴 , 𝑗 𝜖 𝐵

𝑤𝑖𝑗

Graph Cuts

Mincut : min Cut(A1,A2)

However, mincut simply seperates one individual vertex from the rest of the graph

Balanced cut

Problem: finding an optimal graph (normalized) cut is NP-hardApproximation: spectral graph partitioning

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Spectral Clustering

•Unnormalized graph Laplacian

• Normalized graph Laplacian

• Other explanation

• Example

Spectral clustering - main algorithms

Input: Similarity matrix S, number k of clusters to construct• Build similarity graph• Compute the first k eigenvectors v1, . . . , vk of the problem matrix

L for unnormalized spectral clusteringLrw for normalized spectral clustering

• Build the matrix V є Rn×k with the eigenvectors as columns• Interpret the rows of V as new data points Zi є Rk

• Cluster the points Zi with the k-means algorithm in Rk

Example-1

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0 1 1 0 0

1 0 1 0 0

1 1 0 0 0

0 0 0 0 1

0 0 0 1 05

2 0 0 0 0

0 2 0 0 0

0 0 2 0 0

0 0 0 1 0

0 0 0 0 1

L: Laplacian matrix2 -1 -1 0 0

-1 2 -1 0 0

-1 -1 2 0 0

0 0 0 1 -1

0 0 0 -1 1

Similarity Graph

W: adjacency matrix D: degree matrix

Example-1

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Similarity Graph 1 0

1 0

1 0

0 1

0 1

L: Laplacian matrix

2 -1 -1 0 0

-1 2 -1 0 0

-1 -1 2 0 0

0 0 0 1 -1

0 0 0 -1 1

Double Zero Eigenvalue Two Connected Components

First TwoEigenvectors

v1 v2

Example-1

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Similarity Graph

First k Eigenvectors New Clustering Space

1 0

1 0

1 0

0 1

0 1

y1

y2

y3

y4

y5

v1 v2

Use k-means clustering in the new space

v2

v1

Unnormalized graph LaplacianDefine as L=D-W

proof

Unnormalized graph Laplacian

proof

Relation between spectrum and clusters: Multiplicity of k eigenvalue 0 = number k of connectedcomponents A1, ..., Ak of the graph. eigenspace is spanned by the characteristic functions 1A1 , ..., 1Akof those components (so all eigenvecotrs are piecewise constant).

Unnormalized graph Laplacian

Interpret sij = 1 / d(Xi , Xj )2

looks like a discrete version of the standard Laplace operator

Normalized graph Laplacian

Define

Relation between Lsym & Lrw

Lsym Lrw

Eigenvalue λ λ

Eigenvector D1/2u u

Normalized graph Laplacian

Spectral properties similar to L:• Positive semi-definite, smallest eigenvalue is 0• Attention: For Lrw, eigenspace spanned by 1Ai (piecewise const.)but for Lsym, eigenspace spanned by D1/21Ai (not piecewiseconst).

Random walk explanationsGeneral observation:• Random walk on the graph has transition matrix P = D−1W.• note that Lrw = I − P

Specific observation about Ncut :• define P(A|B) is the probability to jump from B to A if weassume that the random walk starts in the stationarydistribution.• Then: Ncut(A, B) = P(A|B) + P(B|A)• Interpretation: Spectral clustering tries to construct groups such that a random walk stays as long as possible within the same group

Possible Explanations

Example-2

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-0.709 -0.7085 -0.708 -0.7075 -0.707 -0.7065 -0.706In the embedded space given by two leading eigenvectors, clusters are trivial to separate.

Example-3

In the embedded space given by three leading eigenvectors, clusters are trivial to separate.

Another point of view

Connections

PCALinear combination the original data Xi to now variable Zi

Rank reduce comparison between PCA & Laplacian Eigenmap

PCA is linear combination to reduce dimension, though PCA minimize the Reconstruction error, but it’s not helpful to cluster groups.Spectral clustering is nonlinear reducing dimension which is helpful to cluster, however it’s doesn’t actually have a “rank reduce function” apply to new data, while PCA have it.

ConclusionWhy is spectral clustering useful?• Does not make strong assumptions on cluster shape• Is simple to implement (solving an eigenproblem)• Spectral clustering objective does not have local optima• Has several different derivations• Successful in many applications

What are potential problems?• Can be sensitive to choice of parameters (k in kNN-graph).• Computational expensive on large non-sparse graphs