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Spectral Clustering

Spectral Clustering

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Spectral Clustering

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  1. Spectral Clustering Jianping Fan Dept of Computer Science UNC, Charlotte

  2. Lecture Outline • Motivation • Graph overview and construction • Spectral Clustering • Cool implementations

  3. Semantic interpretations of clustering clusters

  4. 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. Spectral Clustering Example – 2 Spirals

  5. Spectral Clustering Example Original Points K-means (2 Clusters) Why k-means fail for these two examples?

  6. Lecture Outline • Motivation • Graph overview and construction • Spectral Clustering • Cool implementation

  7. Graph-based Representation of Data Similarity

  8. similarity Graph-based Representation of Data Similarity

  9. Graph-based Representation of Data Relationship

  10. Manifold

  11. Graph-based Representation of Data Relationships Manifold

  12. Graph-based Representation of Data Relationships

  13. Data Graph Construction

  14. Graph-based Representation of Data Relationships

  15. Graph-based Representation of Data Relationships

  16. Graph-based Representation of Data Relationships

  17. Graph-based Representation of Data Relationships

  18. Graph Cut

  19. Lecture Outline • Motivation • Graph overview and construction • Spectral Clustering • Cool implementations

  20. Graph-based Representation of Data Relationships

  21. Graph Cut

  22. Graph-based Representation of Data Relationships

  23. Graph Cut

  24. Eigenvectors & Eigenvalues

  25. Normalized Cut A graph G(V, E) can be partitioned into two disjoint sets A, B Cut is defined as: Optimal partition of the graph G is achieved by minimizing the cut Min ( )

  26. Normalized Cut Normalized Cut Association between partition set and whole graph

  27. Normalized Cut

  28. Normalized Cut

  29. Normalized Cut

  30. Normalized Cut Normalized Cut becomes Normalized cut can be solved by eigenvalue equation:

  31. K-way Min-Max Cut Intra-cluster similarity Inter-cluster similarity Decision function for spectral clustering

  32. Mathematical Description of Spectral Clustering Refined decision function for spectral clustering We can further define:

  33. Refined decision function for spectral clustering This decision function can be solved as

  34. Spectral Clustering Algorithm Ng, Jordan, and Weiss • Motivation • Given a set of points • We would like to cluster them into k subsets

  35. 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

  36. 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

  37. 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

  38. Algorithm • Final Cluster Assignment • Assign point to cluster j iff row i of Y was assigned to cluster j