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אשכול בעזרת אלגורתמים בתורת הגרפים

אשכול בעזרת אלגורתמים בתורת הגרפים. רועי יצחק. Elements of Graph Theory. A 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 vertices

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אשכול בעזרת אלגורתמים בתורת הגרפים

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  1. אשכול בעזרת אלגורתמים בתורת הגרפים רועי יצחק

  2. Elements of Graph Theory • A graphG = (V,E) consists of a vertex setV and an edge setE. • If G is a directed graph, each edge is an ordered pair of vertices • A bipartite graph is one in which the vertices can be divided into two groups, so that all edges join vertices in different groups.

  3. 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 . . . העברת הנק' לגרף • קבע את הנק' במישור • קבע מרחק בין כל זוג נקודות graph representation distance matrix n-D data points

  4. 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

  5. V={xi} Set of n vertices representing data points E={Wij} Set of weighted edges indicating pair-wise similarity between points 0.1 5 0.8 1 0.8 0.8 0.6 6 2 4 0.7 0.8 0.2 3 Similarity Graph • Distance decrease similarty increase • Represent dataset as a weighted graph G(V,E)

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

  7. Graph Partitioning • Clustering can be viewed as partitioning a similarity graph • Bi-partitioning task: • Divide vertices into two disjoint groups (A,B) A B 5 1 2 4 6 3 V=A U B

  8. Apply these objectives to our graph representation 0.1 5 0.8 1 0.8 0.8 0.6 6 2 4 0.7 0.2 0.8 3 Minimize weight of between-group connections Clustering Objectives • Traditional definition of a “good” clustering: • Points assigned to same cluster should be highly similar. • Points assigned to different clusters should be highly dissimilar.

  9. A B 0.1 0.8 5 1 0.8 0.8 cut(A,B) = 0.3 0.6 6 4 2 0.7 0.8 0.2 3 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

  10. Degenerate case: Optimal cut Minimum cut • Problem: • Only considers external cluster connections • Does not consider internal cluster density Graph Cut Criteria • Criterion: Minimum-cut • Minimise weight of connections between groups min cut(A,B)

  11. Graph Cut Criteria (continued) • Criterion: Normalised-cut (Shi & Malik,’97) • Consider the connectivity between groups relative to the density of each group. • 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.

  12. עץ פורש מינימאלי (MST-Prim) • קבע קודקוד מקור והכנס אותו לסט A (עץ) • מצא את הקשת הקלה ביותר אשר המחברת בין הסט B (שאר הקודקודים בגרף) לעץ (A) • חזור על התהליך עד שלא ישארו קודקודים בסט B

  13. מציאת clustring • קבע את כיוון ההתקדמות בעץ (כל הוספה של צומת) בפונקציה של משקל הקשת שהוספה • כל "עמק" בגרף מייצג cluster

  14.     4          8 8 8 7 7 b b c c d d 9 9 4 4 2 2 a a e e i i 11 11 14 14 4 4 6 6 7 7 8 8 10 10 h h g g f f 2 2 1 1 Example

  15. 4    8 8 8 7 7 b b c c d d 8  4 9 9 4 4 2 2 a a e e i i 11 11 14 14 4 4   6 6 7 7 8 8 10 10 h h g g f f 2 2 1 1   8 Example 8   7 2 4

  16. 8 8 7 7 b b c c d d 8 4 7 8 4 7 9 9 4 4 2 2 a a e e i i 11 11 14 14 4 4  2  2 6 6 7 7 8 8 10 10 h h g g f f 2 2 1 1 4  8 4 6 7 Example 6 7 10 2

  17. 8 8 7 7 b b c c d d 8 8 4 4 7 7 9 9 4 4 2 2 a a e e i i 11 11 14 14 4 4 10 10 2 2 6 6 7 7 8 8 10 10 h h g g f f 2 2 1 1 4 4 2 2 1 7 Example 1

  18. 8 7 b c d 8 4 7 9 4 2 a e i 11 14 4 10 2 6 7 8 10 h g f 2 1 4 2 1 Example 9

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

  20. Dataset exhibits complex cluster shapes • K-means performs very poorly in this space due bias toward dense spherical clusters. In the embedded space given by two leading eigenvectors, clusters are trivial to separate. Example – 2 Spirals

  21. The eigenvalues and eigenvectors of a matrix provide global information about its structure. • 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. Spectral Graph Theory • Possible approach • Represent a similarity graph as a matrix • Apply knowledge from Linear Algebra…

  22. Matrix Representations • Adjacency matrix (A) • n x n matrix • : edge weight between vertex xiand xj 0.1 0.8 5 1 0.8 0.8 0.6 6 4 2 0.7 0.8 0.2 3 • Important properties: • Symmetric matrix • Eigenvalues are real • Eigenvector could span orthogonal base

  23. Degree matrix (D) n x n diagonal matrix : total weight of edges incident to vertex xi Matrix Representations (continued) 0.1 0.8 5 1 0.8 0.8 0.6 6 4 2 0.7 0.8 0.2 3 • Important application: • Normalise adjacency matrix

  24. Matrix Representations (continued) L = D - A • Laplacian matrix (L) • n x n symmetric matrix 0.1 0.8 5 1 0.8 0.8 0.6 6 4 2 0.7 0.8 0.2 3 • 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…

  25. Another option – normalized laplasian • Laplacian matrix (L) • n x n symmetric matrix 0.1 0.8 5 1 0.8 0.8 0.6 6 4 2 0.7 0.8 0.2 3 • Important properties: • Eigenvectors are real and normalize • Each Aij which i,j is not equal =

  26. Spectral Clustering Algorithms • Three basic stages: • Pre-processing • Construct a matrix representation of the dataset. • Decomposition • Compute eigenvalues and eigenvectors of the matrix. • Map each point to a lower-dimensional representation based on one or more eigenvectors. • Grouping • Assign points to two or more clusters, based on the new representation.

  27. 0.0 • Decomposition • Find eigenvalues Xand eigenvectors Λof the matrix L 0.4 0.2 0.1 0.4 -0.2 -0.9 0.4 0.4 0.2 0.1 -0. 0.4 0.3 2.2 0.4 0.2 -0.2 0.0 -0.2 0.6 Λ = X = 2.3 0.4 -0.4 0.9 0.2 -0.4 -0.6 0.4 -0.7 -0.4 -0.8 -0.6 -0.2 2.5 0.4 -0.7 -0.2 0.5 0.8 0.9 3.0 x1 0.2 • Map vertices to corresponding components of λ2 x2 0.2 x3 0.2 x4 -0.4 x5 -0.7 x6 -0.7 Spectral Bi-partitioning Algorithm • Pre-processing • Build Laplacian matrix L of the graph

  28. Spectral Bi-partitioning Algorithm The matrix which represents the eigenvector of the laplacian the eigenvector matched to the corresponded eigenvalues with increasing order

  29. Split at 0 Cluster A: Positive points Cluster B: Negative points x1 0.2 B A x2 0.2 x3 0.2 x4 -0.4 x1 0.2 x4 -0.4 x5 -0.7 x2 0.2 x5 -0.7 x6 -0.7 x3 0.2 x6 -0.7 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

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