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Spectral graph theory


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  • 1. Presented by Danushka Bollegala
  • 2.  Spectrum = the set of eigenvalues  By looking at the spectrum we can know about the graph itself!  A way of normalizing data (canonical form) and then perform clustering (e.g. via k- means) on this normalized/reduced space.  Input: A similarity matrix  Output: A set of (non-overlapping/hard) clusters.
  • 3.  UndirectedGraph G(V, E)  V: set of vertices (nodes in the network)  E: set of edges (links in the network) ▪ Weight wij is the weight of the edge connecting vertex I and j (represented by the affinity matrix.)  Degree: sum of weights on outgoing edges of a vertex.  Measuring the size of a subset A ofV
  • 4.  How to create the affinity matrixW from the similarity matrix S?  ε-neighborhood graph ▪ Connect all vertices that have similarity greater than ε  k-nearest neighbor graph ▪ Connect the k-nearest neighbors of each vertex. ▪ Mutual k-nearest neighbor graphs for asymmetric S.  Fully connected graph ▪ Use the Gaussian similarity function (kernel)
  • 5.  L = D –W  D: degree matrix. A diagonal matrix diag(d1,...,dn)  Properties  For every vector  L is symmetric and positive semi-definite  The smallest eigenvalue of L is zero and the corresponding eigenvector is 1 = (1,...,1)T  L has n non-negative, real-valued eigenvalues
  • 6.  Two versions exist  Lsym = D-1/2LD-1/2 = I - D-1/2WD-1/2  Lrw = D-1L = I - D-1W
  • 7.  The partition (A1,...,Ak) induces a cut on the graph  Two types of graph cuts exist  Spectral clustering solves a relaxed version of the mincut problem (therefore it is an approximation)
  • 8. By the Rayleigh-Ritz theorem it follows that the second eigenvalue is the minimum.
  • 9.  Transition probability matrix and Laplacian are related!  P = D-1W  Lrw = I - P
  • 10.  Lrw based spectral clustering (Shi & Malik,2000) is better (especially when the degree distribution is uneven).  Use k-nearest neighbor graphs  How to set the number of clusters:  k=log(n)  Use the eigengap heuristic  If using Gaussian kernel how to set sigma  Mean distance of a point to its log(n)+1 nearest neighbors.
  • 11.  Eckart-YoungTheorem  The low-rank approximation B for a matrix A s.t. rank(B) = r < rank(A) is given by,  B = USV*, where A = UZV* and S is the same as Z except the (r+1) and above singular values of Z are set to zero.  Approximation is done by minimizing the Frobenius norm ▪ minB||A – B||F, subject to rank(B) = r