LecXX5 - constant values in 2 nd eigenvector 4/6/10 COT...

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1 4/6/10 COT 6936 1 COT 6936: Topics in Algorithms Giri Narasimhan ECS 254A / EC 2443; Phone: x3748 giri@cs.fiu.edu http://www.cs.fiu.edu/~giri/teach/COT6936_S10.html https://online.cis.fiu.edu/portal/course/view.php?id=427 Spectral Methods ± Graph Connectivity problems ± Google Page Rank ± Graph Partitioning problems ± Clustering (even linearly non-separable case) ± Markov Chain Mixing problems ± Random walks in graphs 4/6/10 COT 6936 2 Matrices and Eigenvalues ± Array of values ± Linear Transformation ± Eigenvalues and Eigenvectors ± Ax = ± x ± Under transformation A, eigenvectors only experience change in magnitude, not direction ± A = Q ² Q -1 4/6/10 COT 6936 3
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2 Graph Bisection ± Construct adjacency matrix A ± Construct Laplacian L = D – A ± L is positive semi-definite (PSD); has non-neg eigenvalues; has smallest eigenvalue = 0 ± Second eigenvector provides information about bisection. ± Signs of 2 nd eigenvector give a good bisection ± Extreme case : Connected components have
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Unformatted text preview: constant values in 2 nd eigenvector 4/6/10 COT 6936 4 Graph Bisection (Continued) Eigenvalues indicate strength of bisection How to get bisections with n/2 vertices? Use median value in second eigenvector How to get k partitions? Perform bisections recursively Use more eigenvectors 4/6/10 COT 6936 5 Spectral Clustering: Strategy Given data points and a distance function, construct a weighted graph Let A be its adjacency matrix; let D be diagonal matrix with degrees along diagonal Construct Laplacian L ( PSD , non-neg eigenv .) Unnormalized: L = D A Normalized symmetric: L = D-1/2 LD 1/2 Random Walk: L = D-1 L Matrix L k has cols = first k eigenvectors of L Cluster rows of L k . 4/6/10 COT 6936 6 3 Spectral Clustering Need distance measure (need not be a metric), i.e., triangle inequality not needed Not Model-based Global method Turns discrete problem into continuous 4/6/10 COT 6936 7...
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LecXX5 - constant values in 2 nd eigenvector 4/6/10 COT...

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