Generalized median graph computation by means of graph embedding in vector spaces

The median graph has been presented as a useful tool to represent a set of graphs. Nevertheless its computation is very complex and the existing algorithms are restricted to use limited amount of data. In this paper we propose a new approach for the computation of the median graph based on graph embedding. Graphs are embedded into a vector space and the median is computed in the vector domain. We have designed a procedure based on the weighted mean of a pair of graphs to go from the vector domain back to the graph domain in order to obtain a final approximation of the median graph. Experiments on three different databases containing large graphs show that we succeed to compute good approximations of the median graph. We have also applied the median graph to perform some basic classification tasks achieving reasonable good results. These experiments on real data open the door to the application of the median graph to a number of more complex machine learning algorithms where a representative of a set of graphs is needed.

[1]  Abraham Kandel,et al.  Graph-Theoretic Techniques for Web Content Mining , 2005, Series in Machine Perception and Artificial Intelligence.

[2]  Edwin R. Hancock,et al.  A Riemannian approach to graph embedding , 2007, Pattern Recognit..

[3]  Kaspar Riesen,et al.  Approximate graph edit distance computation by means of bipartite graph matching , 2009, Image Vis. Comput..

[4]  Kaspar Riesen,et al.  Graph Embedding in Vector Spaces by Means of Prototype Selection , 2007, GbRPR.

[5]  Kaspar Riesen,et al.  Fast Suboptimal Algorithms for the Computation of Graph Edit Distance , 2006, SSPR/SPR.

[6]  Shigeo Abe DrEng.  Pattern Classification , 2001, Springer London.

[7]  Alberto Sanfeliu,et al.  Synthesis of Median Spectral Graph , 2005, IbPRIA.

[8]  Ernest Valveny,et al.  Report on the Second Symbol Recognition Contest , 2005, GREC.

[9]  Chandrajit L. Bajaj,et al.  The algebraic degree of geometric optimization problems , 1988, Discret. Comput. Geom..

[10]  M. Shirosaki.  Another proof of the defect relation for moving targets , 1991 .

[11]  Edwin R. Hancock,et al.  Spectral embedding of graphs , 2003, Pattern Recognit..

[12]  Mario Vento,et al.  Thirty Years Of Graph Matching In Pattern Recognition , 2004, Int. J. Pattern Recognit. Artif. Intell..

[13]  Kaspar Riesen,et al.  IAM Graph Database Repository for Graph Based Pattern Recognition and Machine Learning , 2008, SSPR/SPR.

[14]  Shengrui Wang,et al.  Median graph computation for graph clustering , 2006, Soft Comput..

[15]  Horst Bunke,et al.  Weighted Mean of a Pair of Graphs , 2001, Computing.

[16]  King-Sun Fu,et al.  A distance measure between attributed relational graphs for pattern recognition , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[17]  David G. Stork,et al.  Pattern Classification , 1973 .

[18]  Horst Bunke,et al.  Inexact graph matching for structural pattern recognition , 1983, Pattern Recognit. Lett..

[19]  Slobodan Ribarić,et al.  Introduction to Pattern Recognition , 1988 .

[20]  Robert P. W. Duin,et al.  Prototype selection for dissimilarity-based classifiers , 2006, Pattern Recognit..

[21]  Piotr Indyk,et al.  Algorithmic applications of low-distortion geometric embeddings , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[22]  Edwin R. Hancock,et al.  Pattern Vectors from Algebraic Graph Theory , 2005, IEEE Trans. Pattern Anal. Mach. Intell..

[23]  Anil K. Jain,et al.  Introduction to Pattern Recognition , 2007 .

[24]  Horst Bunke,et al.  An Error-Tolerant Approximate Matching Algorithm for Attributed Planar Graphs and Its Application to Fingerprint Classification , 2004, SSPR/SPR.

[25]  Horst Bunke,et al.  On Median Graphs: Properties, Algorithms, and Applications , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[26]  Francisco Casacuberta,et al.  Topology of Strings: Median String is NP-Complete , 1999, Theor. Comput. Sci..

[27]  Alfred O. Hero,et al.  A binary linear programming formulation of the graph edit distance , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[28]  Richard C. Wilson,et al.  Mixing spectral representations of graphs , 2006, 18th International Conference on Pattern Recognition (ICPR'06).

[29]  M. Fatih Demirci,et al.  Object Recognition as Many-to-Many Feature Matching , 2006, International Journal of Computer Vision.

[30]  Trevor Darrell,et al.  Fast contour matching using approximate earth mover's distance , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..