Ranking and Sparsifying a Connection Graph

Many problems arising in dealing with high-dimensional data sets involve connection graphs in which each edge is associated with both an edge weight and a d-dimensional linear transformation. We consider vectorized versions of the PageRank and effective resistance which can be used as basic tools for organizing and analyzing complex data sets. For example, the generalized PageRank and effective resistance can be utilized to derive and modify diffusion distances for vector diffusion maps in data and image processing. Furthermore, the edge ranking of the connection graphs determined by the vectorized PageRank and effective resistance are an essential part of sparsification algorithms which simplify and preserve the global structure of connection graphs.

[1]  K. Markström,et al.  Expansion properties of random Cayley graphs and vertex transitive graphs via matrix martingales , 2008 .

[2]  I. Jolliffe.  Principal Component Analysis , 2002 .

[3]  David R. Karger,et al.  Using randomized sparsification to approximate minimum cuts , 1994, SODA '94.

[4]  Shang-Hua Teng,et al.  Spectral Sparsification of Graphs , 2008, SIAM J. Comput..

[5]  David R. Karger,et al.  Minimum cuts in near-linear time , 1998, JACM.

[6]  A. Singer.  Angular Synchronization by Eigenvectors and Semidefinite Programming. , 2009, Applied and computational harmonic analysis.

[7]  Dimitris Achlioptas,et al.  Database-friendly random projections , 2001, PODS.

[8]  Richard Szeliski,et al.  Building Rome in a day , 2009, ICCV.

[9]  A. Singer,et al.  Representation theoretic patterns in three dimensional Cryo-Electron Microscopy I: The intrinsic reconstitution algorithm. , 2009, Annals of mathematics.

[10]  Nikhil Srivastava,et al.  Graph sparsification by effective resistances , 2008, SIAM J. Comput..

[11]  Benjamin Recht,et al.  A Simpler Approach to Matrix Completion , 2009, J. Mach. Learn. Res..

[12]  Fan Chung Graham,et al.  Laplacian and vibrational spectra for homogeneous graphs , 1992, J. Graph Theory.

[13]  G. Kirchhoff.  Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird , 1847 .

[14]  Joel A. Tropp,et al.  User-Friendly Tail Bounds for Sums of Random Matrices , 2010, Found. Comput. Math..

[15]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[16]  François Fouss,et al.  Random-Walk Computation of Similarities between Nodes of a Graph with Application to Collaborative Recommendation , 2007, IEEE Transactions on Knowledge and Data Engineering.

[17]  Sergey Brin,et al.  The Anatomy of a Large-Scale Hypertextual Web Search Engine , 1998, Comput. Networks.

[18]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[19]  Fan Chung Graham,et al.  Ranking and Sparsifying a Connection Graph , 2014, Internet Math..

[20]  Mark Herbster,et al.  Fast Prediction on a Tree , 2008, NIPS.

[21]  Pavel Berkhin,et al.  Bookmark-Coloring Algorithm for Personalized PageRank Computing , 2006, Internet Math..

[22]  Yaron Lipman,et al.  Sensor network localization by eigenvector synchronization over the euclidean group , 2012, TOSN.

[23]  Van H. Vu,et al.  Spectral norm of random matrices , 2007, Comb..

[24]  David R. Karger,et al.  Approximating s – t Minimum Cuts in ~ O(n 2 ) Time , 2007 .

[25]  Mustafa Yilmaz,et al.  Genetic clustering of social networks using random walks , 2007, Comput. Stat. Data Anal..

[26]  Fan Chung Graham,et al.  Local Graph Partitioning using PageRank Vectors , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[27]  A. Guionnet,et al.  An Introduction to Random Matrices , 2009 .

[28]  Fan Chung Graham,et al.  On the Spectra of General Random Graphs , 2011, Electron. J. Comb..

[29]  Shang-Hua Teng,et al.  A Sublinear Time Algorithm for PageRank Computations , 2012, WAW.

[30]  David R. Karger,et al.  Random Sampling in Cut, Flow, and Network Design Problems , 1999, Math. Oper. Res..

[31]  Gary L. Miller,et al.  Approaching Optimality for Solving SDD Linear Systems , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[32]  Amit Singer,et al.  A Cheeger Inequality for the Graph Connection Laplacian , 2012, SIAM J. Matrix Anal. Appl..

[33]  David G. Lowe,et al.  Object recognition from local scale-invariant features , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[34]  Shang-Hua Teng,et al.  Nearly-Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems , 2006, SIAM J. Matrix Anal. Appl..

[35]  Avi Wigderson,et al.  Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators, and applications , 2008, Theory Comput..

[36]  Zhizhen Zhao,et al.  Viewing Angle Classification of Cryo-Electron Microscopy Images Using Eigenvectors , 2011, SIAM J. Imaging Sci..

[37]  Shang-Hua Teng,et al.  Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems , 2003, STOC '04.

[38]  Fan Chung Graham,et al.  A Sharp PageRank Algorithm with Applications to Edge Ranking and Graph Sparsification , 2010, WAW.

[39]  A. Singer,et al.  Vector diffusion maps and the connection Laplacian , 2011, Communications on pure and applied mathematics.

[40]  R. Penrose.  A Generalized inverse for matrices , 1955 .

[41]  Jennifer Widom,et al.  Scaling personalized web search , 2003, WWW '03.