Weighted Theta Functions and Embeddings with Applications to Max-Cut, Clustering and Summarization

We introduce a unifying generalization of the Lovasz theta function, and the associated geometric embedding, for graphs with weights on both nodes and edges. We show how it can be computed exactly by semidefinite programming, and how to approximate it using SVM computations. We show how the theta function can be interpreted as a measure of diversity in graphs and use this idea, and the graph embedding in algorithms for Max-Cut, correlation clustering and document summarization, all of which are well represented as problems on weighted graphs.

[1]  Avrim Blum,et al.  Correlation Clustering , 2004, Machine Learning.

[2]  Shih-Fu Chang,et al.  Semi-supervised learning using greedy max-cut , 2013, J. Mach. Learn. Res..

[3]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[4]  Abraham Duarte,et al.  Advanced Scatter Search for the Max-Cut Problem , 2009, INFORMS J. Comput..

[5]  Hui Lin,et al.  A Class of Submodular Functions for Document Summarization , 2011, ACL.

[6]  Sanjeev Arora,et al.  Fast algorithms for approximate semidefinite programming using the multiplicative weights update method , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[7]  Donald E. Knuth.  The Sandwich Theorem , 1994, Electron. J. Comb..

[8]  Alexander Schrijver,et al.  A comparison of the Delsarte and Lovász bounds , 1979, IEEE Trans. Inf. Theory.

[9]  Michel X. Goemans,et al.  Semideenite Programming in Combinatorial Optimization , 1999 .

[10]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[11]  D. Pham,et al.  Selection of K in K-means clustering , 2005 .

[12]  Aristides Gionis,et al.  Overlapping Correlation Clustering , 2011, ICDM.

[13]  Clifford Stein,et al.  Approximating Semidefinite Packing Programs , 2011, SIAM J. Optim..

[14]  Franz Rendl,et al.  A Spectral Bundle Method for Semidefinite Programming , 1999, SIAM J. Optim..

[15]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[16]  Devdatt P. Dubhashi,et al.  Lovász ϑ function, SVMs and finding dense subgraphs , 2013, J. Mach. Learn. Res..

[17]  Sanjeev Arora,et al.  The Multiplicative Weights Update Method: a Meta-Algorithm and Applications , 2012, Theory Comput..

[18]  L. Lovász.  Geometric representations of graphs , 2014 .

[19]  M. Elsner,et al.  Bounding and Comparing Methods for Correlation Clustering Beyond ILP , 2009, ILP 2009.

[20]  E. D. Klerk,et al.  The Lovász ϑ-Function , 2002 .

[21]  R. Monteiro,et al.  A projected gradient algorithm for solving the maxcut SDP relaxation , 2001 .

[22]  Don R. Hush,et al.  QP Algorithms with Guaranteed Accuracy and Run Time for Support Vector Machines , 2006, J. Mach. Learn. Res..

[23]  M. Brand,et al.  Fast low-rank modifications of the thin singular value decomposition , 2006 .

[24]  Devdatt P. Dubhashi,et al.  Lovasz ϑ, SVMs and applications , 2013, 2013 IEEE Information Theory Workshop (ITW).

[25]  Bernhard Schölkopf,et al.  Estimating the Support of a High-Dimensional Distribution , 2001, Neural Computation.