Spectral Clustering Eyal David Image Processing seminar May 2008

Spectral Clustering

Eyal David

Image Processing seminar

May 2008

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Lecture Outline

Motivation Graph overview and construction Demo Spectral Clustering Demo Cool implementations

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A Tutorial on Spectral Clustering\Arik Azran

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Spectral Clustering Example – 2 Spirals

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Dataset exhibits complex Dataset exhibits complex cluster shapescluster shapes

K-means performs very K-means performs very poorly in this space due poorly in this space due bias toward dense bias toward dense spherical clusters.spherical clusters.

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

Spectral Clustering - Derek Greene

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Lecture Outline

Motivation Graph overview and construction Graph demo Spectral Clustering Spectral Clustering demo Cool implementation

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Matthias Hein and Ulrike von Luxburg August 2007

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Demo

(Live example)

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Lecture Outline

Motivation Graph overview and construction Demo Spectral Clustering Demo Cool implementations

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Eigenvectors & Eigenvalues

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Demo

(Live example)

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Spectral Clustering Algorithm Ng, Jordan, and Weiss

Motivation Given a set of points

We would like to cluster them into k subsets

1,...,l

nS s s R

Slides from Spectral Clustering by Rebecca Nugent, Larissa Stanberry based on Ng et al On Spectral clustering: analysis and algorithm

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Algorithm

Form the affinity matrix Define if

Scaling parameter chosen by user

Define D a diagonal matrix whose

(i,i) element is the sum of A’s row i

nxnW Ri j

0iiW

2 2|| || / 2i js s

ijW e


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Algorithm

Form the matrix

Find , the k largest eigenvectors of L These form the the columns of the new

matrix X Note: have reduced dimension from nxn to nxk

1/ 2 1/ 2L D WD

1 2, ,..., kx x x


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Algorithm

Form the matrix Y Renormalize each of X’s rows to have unit length Y

Treat each row of Y as a point in Cluster into k clusters via K-means

2 2/( )ij ij ijj

Y X X kR

nxkR


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Algorithm

Final Cluster Assignment Assign point to cluster j iff row i of Y was

assigned to cluster jis


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Why?

If we eventually use K-means, why not just apply K-means to the original data?

This method allows us to cluster non-convex regions


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Some Examples

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Ng et al On Spectral clustering: analysis and algorithm

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User’s Prerogative

Affinity matrix construction Choice of scaling factor

Realistically, search over and pick value that gives the tightest clusters

Choice of k, the number of clusters Choice of clustering method

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K

Eig

enva

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Largest Largest eigenvalueseigenvalues

of Cisi/Medline of Cisi/Medline datadata

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How to select k? Eigengap: the difference between two consecutive eigenvalues. Most stable clustering is generally given by the value k that

maximises the expression

1k k k

Choose Choose k=2k=2

12max k

Spectral Clustering - Derek Greene

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Recap – The bottom line

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Summary

Spectral clustering can help us in hard clustering problems

The technique is simple to understand The solution comes from solving a simple

algebra problem which is not hard to implement

Great care should be taken in choosing the “starting conditions”

The End

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Spectral Clustering Eyal David Image Processing seminar May 2008