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- 1. Notes on Spectral Clustering Politecnico di Milano, 28/05/2012 Davide Eynard Institute of Computational Sciences - Faculty of Informatics Università della Svizzera Italiana davide.eynard@usi.ch
- 2. Talk outline Spectral Clustering Distances and similarity graphs Graph Laplacians and their properties Spectral clustering algorithms SC under the hood24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 2/15
- 3. Similarity graph The objective of a clustering algorithm is partitioning data into groups such that: Points in the same group are similar Points in different groups are dissimilar Similarity graph G=(V,E) (undirected graph) Vertices vi and vj are connected by a weighted edge if their similarity is above a given threshold GOAL: find a partition of the graph such that: edges within a group have high weights edges across different groups have low weights24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 3/15
- 4. Weighted adjacency matrix Let G(V,E) be an undirected graph with vertex set V={v1,...,vn} Weighted adjacency matrix W=(wij )i,j=1,...,n wij≥0 is the weight of the edge between vi and vj wij=0 means that vi and vj are not connected by an edge wij=wji Degree of a vertex vi∈V: di=∑j=1..nwij Degree matrix D=diag(d1,...,dn)24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 4/15
- 5. Different similarity graphs ε-neighborhood Connect all points whose pairwise distance is less than ε k-nearest neighbors if vi ∈ knn(vj) OR vj ∈ knn(vi) if vi ∈ knn(vj) AND vj ∈ knn(vi) (mutual knn) after connecting edges, use similarity as weight fully connected all points with similarity sij>0 are connected To control neighborhoods to be local, use a similarity function like the Gaussian: s(xi,xj)=exp(-║xi-xj║2/(2σ2))24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 5/15
- 6. Graph Laplacians Graph Laplacian: L=D–W (symmetric and positive semi-definite) Properties Smallest eigenvalue λ1=0 with eigenvector = � n non-negative, real-valued eigenvalues 0=λ1≤λ2≤...≤λn the multiplicty k of the eigenvalue 0 of L equals the number of connected components A1,...,Ak in the graph24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 6/15
- 7. Spectral Clustering algorithm (1) Spectral Clustering algorithm Input: Similarity matrix S ∈ ℝn×n, number k of clusters to construct. 1. Construct a similarity graph as previously described. Let W be its weighted adjacency matrix. 2. Compute the unnormalized Laplacian L 3. Compute the first k eigenvectors u1,…,uk of L 4. Let U ∈ ℝn×k be the matrix containing the vectors u1,…,uk as columns 5. For i=1,...,n let yi ∈ ℝk be the vector corresponding to the i-th row of U 6. Cluster the points (yi)i=1,...,n in ℝk with the k-means algorithm into clusters C1,...,Ck. Output: Clusters A1,...,Ak with Ai= {j | yj ∈ Ci}.24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 7/15
- 8. Normalized Graph Laplacians Normalized graph Laplacians Symmetric: Lsym=D-1/2LD-1/2 = I-D-1/2WD-1/2 Random Walk: Lrw=D-1L=I-D-1W Properties λ is an eigenvalue of Lrw with eigenvector u iff λ is an eigenvalue of Lsym with eigenvector w=D1/2u λ is an eigenvalue of Lrw with eigenvector u iff λ and u solve the generalized eigenproblem Lu=λDu24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 8/15
- 9. Normalized Graph Laplacians Normalized graph Laplacians Symmetric: Lsym=D-1/2LD-1/2 = I-D-1/2WD-1/2 Random Walk: Lrw=D-1L=I-D-1W Properties (follow) 0 is an eigenvalue of Lrw with � as eigenvector, and an eigenvalue of Lsym with eigenvector D1/2�. Lsym and Lrw are positive semi-definite and have n non- negative, real-valued eigenvalues 0=λ1≤λ2≤...≤λn the multiplicty k of the eigenvalue 0 of both Lsym and Lrw equals the number of connected components A1,...,Ak24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 9/15
- 10. Spectral Clustering algorithm (2) Normalized Spectral Clustering Input: Similarity matrix S ∈ ℝn×n, number k of clusters to construct. Lrw: 3. Compute the first k generalized eigenvectors u1,…,uk of the generalized eigenproblem Lu=λDu Lsym: 2. Compute the normalized Laplacian Lsym 3. Compute the first k eigenvectors u1,…,uk of Lsym 4. normalize the eigenvectors Output: Clusters A1,...,Ak with Ai= {j | yj ∈ Ci}.24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 10/15
- 11. A spectral clustering example24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 11/15
- 12. Under the hood 0-eigenvalues in the ideal case parameters are crucial: k in k nearest neighbors σ in Gaussian kernel k (another one!) in k-means24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 12/15
- 13. Random Walk point of view Random walk: stochastic process which randomly jumps from one vertex to another Clustering: finding a partition such that a random walk stays long within a cluster and seldom jumps between clusters Transition probability pij:=wij/di Transition matrix: P = D-1W => Lrw= I - P λ is an eigenvalue of Lrw with eigenvector u iff 1-λ is an eigenvalue of P with eigenvector u24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 13/15
- 14. References Von Luxburg, U. (2007). A tutorial on spectral clustering. Statistics and Computing, 17(4), 395-416. Springer.24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 14/15
- 15. Thank you! Thanks for your attention! Questions?24/04/2011 Notes on Spectral Clustering and Joint Diagonalization 15/15

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