אשכול בעזרת אלגורתמים בתורת

הגרפיםרועי יצחק

Elements of Graph TheoryA graph G = (V,E) consists of a vertex set V and

an edge set E. If G is a directed graph, each edge is an ordered

pair of verticesA bipartite graph is one in which the vertices can

be divided into two groups, so that all edges join vertices in different groups.

העברת הנק' לגרףקבע את הנק' במישורקבע מרחק בין כל זוג נקודות

0 1 1.5 2 5 6 7 9

1 0 2 1 6.5 6 8 8

1.5 2 0 1 4 4 6 5.5

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n-D data pointsgraph

representationdistance matrix

Minimal Cut Create a graph G(V,E) from the data Compute all the distances in the graph The weight of each edge is the distance Remove edges from the graph according to

some threshold

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E={Wij} Set of weighted edges indicating pair-wise similarity between points

Similarity Graph Distance decrease similarty increase Represent dataset as a weighted graph G(V,E)

1 2 6{ , ,..., }v v v

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V={xi} Set of n vertices representing data points

Similarity Graph Wij represent similarity between vertex If Wij=0 where isn’t similarity Wii=0

Graph Partitioning Clustering can be viewed as

partitioning a similarity graph

Bi-partitioning task: Divide vertices into two disjoint

groups (A,B) 1

23

4

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A B

V=A U B

Clustering Objectives Traditional definition of a “good” clustering:

1. Points assigned to same cluster should be highly similar.2. Points assigned to different clusters should be highly dissimilar.

Minimize weight of between-group connections

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Apply these objectives to our graph representation

Graph Cuts Express partitioning objectives as a function of the

“edge cut” of the partition. Cut: Set of edges with only one vertex in a group.we

wants to find the minimal cut beetween groups. The groups that has the minimal cut would be the partition

BjAi

ijwBAcut,

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A B

cut(A,B) = 0.3

Graph Cut Criteria Criterion: Minimum-cut

Minimise weight of connections between groupsmin cut(A,B)

Optimal cutMinimum cut

Problem: Only considers external cluster connections Does not consider internal cluster density

Degenerate case:

Graph Cut Criteria (continued)

Criterion: Normalised-cut (Shi & Malik,’97) Consider the connectivity between groups

relative to the density of each group.

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)(),(),(min

BvolBAcut

AvolBAcutBANcut

Normalise the association between groups by volume. Vol(A): The total weight of the edges originating from

group A. Why use this criterion?

Minimising the normalised cut is equivalent to maximising normalised association. Produces more balanced partitions.

(MST-Prim)עץ פורש מינימאלי קבע קודקוד מקור והכנס אותו לסטA)עץ( מצא את הקשת הקלה ביותר אשר המחברת בין הסטB שאר הקודקודים(

(A)בגרף( לעץ חזור על התהליך עד שלא ישארו קודקודים בסטB

clustringמציאת קבע את כיוון ההתקדמות בעץ )כל הוספה של צומת( בפונקציה של משקל

הקשת שהוספה בגרף מייצג "עמק"כל cluster

Example

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Example

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Example

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Example

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Example

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Similarity Graph Wij represent similarity between vertex If Wij=0 where isn’t similarity Wii=0

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 Graph Theory Possible approach

Represent a similarity graph as a matrix Apply knowledge from Linear Algebra…

Spectral Graph Theory Analyse the “spectrum” of matrix representing a graph. Spectrum : The eigenvectors of a graph, ordered by the

magnitude(strength) of their corresponding eigenvalues.

},...,,{ 21 n

The eigenvalues and eigenvectors of a matrix provide global information about its structure.

11 1 1 1

1

n

n nn n n

w w x xλ

w w x x

Matrix Representations Adjacency matrix (A)

n x n matrix : edge weight between vertex xi and xj

x1 x2 x3 x4 x5 x6

x1 0 0.8 0.6 0 0.1 0

x2 0.8 0 0.8 0 0 0

x3 0.6 0.8 0 0.2 0 0

x4 0 0 0.2 0 0.8 0.7

x5 0.1 0 0 0.8 0 0.8

x6 0 0 0 0.7 0.8 0

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Important properties: Symmetric matrix Eigenvalues are real Eigenvector could span orthogonal base

][ ijwA

Matrix Representations (continued)

Important application: Normalise adjacency matrix

Degree matrix (D) n x n diagonal matrix : total weight of edges incident to vertex xi

x1 x2 x3 x4 x5 x6

x1 1.5 0 0 0 0 0

x2 0 1.6 0 0 0 0

x3 0 0 1.6 0 0 0

x4 0 0 0 1.7 0 0

x5 0 0 0 0 1.7 0

x6 0 0 0 0 0 1.5

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ijwiiD ),(

Matrix Representations (continued)

Laplacian matrix (L) n x n symmetric matrix

Important properties: Eigenvalues are non-negative real numbers Eigenvectors are real and orthogonal Eigenvalues and eigenvectors provide an insight into the

connectivity of the graph…

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L = D - Ax1 x2 x3 x4 x5 x6

x1 1.5 -0.8 -0.6 0 -0.1 0

x2 -0.8 1.6 -0.8 0 0 0

x3 -0.6 -0.8 1.6 -0.2 0 0

x4 0 0 -0.2 1.7 -0.8 -0.7

x5 -0.1 0 0 0.8- 1.7 -0.8

x6 0 0 0 -0.7 -0.8 1.5

Another option – normalized laplasian Laplacian matrix (L)

n x n symmetric matrix1.00 -0.52 -0.39 0.00 -0.06 0.00

-0.52 1.00 -0.50 0.00 0.00 0.00

-0.39 -0.50 1.00 -0.12 0.00 0.00

0.00 0.00 -0.12 1.00 -0.47 -0.44

-0.06 0.00 0.00 0.47- 1.00 -0.50

0.00 0.00 0.00 -0.44 -0.50 1.00 Important properties: Eigenvectors are real and normalize Each Aij which i,j is not equal =

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ijADii

Spectral Clustering Algorithms Three basic stages:

1. Pre-processing Construct a matrix representation of the dataset.

2. Decomposition Compute eigenvalues and eigenvectors of the matrix. Map each point to a lower-dimensional representation

based on one or more eigenvectors.

3. Grouping Assign points to two or more clusters, based on the

new representation.

Spectral Bi-partitioning Algorithm

1. Pre-processing Build Laplacian

matrix L of the graph

0.90.80.5-0.2-0.70.4

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0.30.4-0.0.10.20.4

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ΛΛ = = XX = =

2. Decomposition Find eigenvalues X

and eigenvectors Λ of the matrix L

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0.2x2

0.2x1 Map vertices to corresponding components of λ2

x1 x2 x3 x4 x5 x6x1 1.5 -0.8 -0.6 0 -0.1 0

x2 -0.8 1.6 -0.8 0 0 0

x3 -0.6 -0.8 1.6 -0.2 0 0

x4 0 0 -0.2 1.7 -0.8 -0.7

x5 -0.1 0 0 -0.8 1.7 -0.8

x6 0 0 0 -0.7 -0.8 1.5

Spectral Bi-partitioning Algorithm

123456

0.41 -0.41 -0.65 -0.31 -0.38 0.11

0.41 -0.44 0.01 0.30 0.71 0.22

0.41 -0.37 0.64 0.04 -0.39 -0.37

0.41 0.37 0.34 -0.45 0.00 0.61

0.41 0.41 -0.17 -0.30 0.35 -0.65

0.41 0.45 -0.18 0.72 -0.29 0.09

The matrix which represents the eigenvector of the laplacian the eigenvector matched to the corresponded eigenvalues with increasing order

Spectral Bi-partitioning (continued) Grouping

Sort components of reduced 1-dimensional vector. Identify clusters by splitting the sorted vector in two.

How to choose a splitting point? Naïve approaches:

Split at 0, mean or median value More expensive approaches

Attempt to minimise normalised cut criterion in 1-dimension

-0.7x6

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0.2x3

0.2x2

0.2x1 Split at 0Split at 0Cluster Cluster AA: Positive points: Positive points

Cluster Cluster BB: Negative : Negative pointspoints

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A B