lecture22-annotated - Machine Learning 10-701/15-781, Fall...

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1 Eric Xing © Eric Xing @ CMU, 2006-2008 1 Machine Learning Machine Learning 10 10 -701/15 701/15 -781, Fall 2008 781, Fall 2008 Spectral Clustering Spectral Clustering Eric Xing Eric Xing Lecture 22, December 1, 2008 Reading: Eric Xing © Eric Xing @ CMU, 2006-2008 2 Data Clustering
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2 Eric Xing © Eric Xing @ CMU, 2006-2008 3 Eric Xing © Eric Xing @ CMU, 2006-2008 4 Data Clustering Compactness Connectivity z Two different criteria z Compactness, e.g., k-means, mixture models z Connectivity, e.g., spectral clustering
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3 Eric Xing © Eric Xing @ CMU, 2006-2008 5 Spectral Clustering Data Similarities Eric Xing © Eric Xing @ CMU, 2006-2008 6 z Some graph terminology z Objects (e.g., pixels, data points) i I = vertices of graph G z Edges ( ij ) = pixel pairs with W ij > 0 z Similarity matrix W = [ W ij ] z Degree d i = Σ j G W ij d A = Σ i A d i degree of A G z Assoc(A,B) = Σ i A Σ j B W ij W ij i j i A A B Weighted Graph Partitioning
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4 Eric Xing © Eric Xing @ CMU, 2006-2008 7 z (edge) cut = set of edges whose removal makes a graph disconnected z weight of a cut: cut( A , B ) = Σ i A Σ j B W ij =Assoc(A,B) z Normalized Cut criteria: minimum cut(A, Ā ) More generally: Cuts in a Graph A A d A A d A A A A ) , ( cut ) , ( cut ) , Ncut( + = = = = = k r A r r k r V j A i ij A V j A i ij k r r r r d A A W W A A A 1 1 2 1 ) , ( cut ) , Ncut( , \ , K Eric Xing © Eric Xing @ CMU, 2006-2008 8 Graph-based Clustering z Data Grouping z Image sigmentation z Affinity matrix: z Degree matrix: z Laplacian matrix: z (bipartite) partition vector: ij W G = {V,E} W ij i j )) , ( ( j i ij x x d f W = ] [ , j i w W = ) ( diag i d D = W D L = ] , , , [ ] [ 1 1 1 1 1 1 1 = = K K , , ,...,x x x N
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5 Eric Xing © Eric Xing @ CMU, 2006-2008 9 Affinity Function z Affinities grow as σ grows Æ z How the choice of σ value affects the results? z What would be the optimal choice for σ ? 2 2 2 σ j i X X j i e W = , Eric Xing Clustering via Optimizing Normalized Cut z The normalized cut: z Computing an optimal normalized cut over all possible y (i.e., partition) is NP hard z
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lecture22-annotated - Machine Learning 10-701/15-781, Fall...

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