Graph Classification by Means of Lipschitz Embedding

In pattern recognition and related fields, graph-based representations offer a versatile alternative to the widely used feature vectors. Therefore, an emerging trend of representing objects by graphs can be observed. This trend is intensified by the development of novel approaches in graph-based machine learning, such as graph kernels or graph-embedding techniques. These procedures overcome a major drawback of graphs, which consists of a serious lack of algorithms for classification. This paper is inspired by the idea of representing graphs through dissimilarities and extends our previous work to the more general setting of Lipschitz embeddings. In an experimental evaluation, we empirically confirm that classifiers that rely on the original graph distances can be outperformed by a classification system using the Lipschitz embedded graphs.

[1]  W. B. Johnson,et al.  Extensions of Lipschitz mappings into Hilbert space , 1984 .

[2]  J. MacQueen.  Some methods for classification and analysis of multivariate observations , 1967 .

[3]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[4]  J. Munkres.  ALGORITHMS FOR THE ASSIGNMENT AND TRANSIORTATION tROBLEMS* , 1957 .

[5]  J. Kazius,et al.  Derivation and validation of toxicophores for mutagenicity prediction. , 2005, Journal of medicinal chemistry.

[6]  Kaspar Riesen,et al.  On Lipschitz Embeddings of Graphs , 2008, KES.

[7]  Radford M. Neal.  Pattern Recognition and Machine Learning , 2007, Technometrics.

[8]  Christopher M. Bishop,et al.  Pattern Recognition and Machine Learning (Information Science and Statistics) , 2006 .

[9]  Ali Shokoufandeh,et al.  Indexing hierarchical structures using graph spectra , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

[11]  Celso C. Ribeiro,et al.  A Randomized Heuristic for Scene Recognition by Graph Matching , 2004, WEA.

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

[13]  Hanan Samet,et al.  Properties of Embedding Methods for Similarity Searching in Metric Spaces , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

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

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

[16]  Terry Caelli,et al.  Inexact Graph Matching Using Eigen-Subspace Projection Clustering , 2004, Int. J. Pattern Recognit. Artif. Intell..

[17]  Myron Wish,et al.  Three-Way Multidimensional Scaling , 1978 .

[18]  Bernard Haasdonk,et al.  Feature space interpretation of SVMs with indefinite kernels , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[19]  H. Hotelling.  Analysis of a complex of statistical variables into principal components. , 1933 .

[20]  Robert P. W. Duin,et al.  The Dissimilarity Representation for Pattern Recognition - Foundations and Applications , 2005, Series in Machine Perception and Artificial Intelligence.

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

[22]  Shinji Umeyama,et al.  An Eigendecomposition Approach to Weighted Graph Matching Problems , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

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

[24]  Remco C. Veltkamp,et al.  Efficient image retrieval through vantage objects , 1999, Pattern Recognit..

[25]  H. Gabriela,et al.  Cluster-preserving Embedding of Proteins , 1999 .

[26]  S. V. N. Vishwanathan,et al.  Graph kernels , 2007 .

[27]  H. Kuhn.  The Hungarian method for the assignment problem , 1955 .

[28]  Horst Bunke,et al.  Transforming Strings to Vector Spaces Using Prototype Selection , 2006, SSPR/SPR.

[29]  Nello Cristianini,et al.  Kernel Methods for Pattern Analysis , 2004 .

[30]  Richard C. Wilson,et al.  Levenshtein distance for graph spectral features , 2004, ICPR 2004.

[31]  András Faragó,et al.  Fast Nearest-Neighbor Search in Dissimilarity Spaces , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[32]  David G. Stork,et al.  Pattern Classification (2nd ed.) , 1999 .

[33]  Kaspar Riesen,et al.  Graph Classification Based on Vector Space Embedding , 2009, Int. J. Pattern Recognit. Artif. Intell..

[34]  Ali S. Hadi,et al.  Finding Groups in Data: An Introduction to Chster Analysis , 1991 .

[35]  Michael J. Fischer,et al.  The String-to-String Correction Problem , 1974, JACM.

[36]  Horst Bunke,et al.  Automatic learning of cost functions for graph edit distance , 2007, Inf. Sci..

[37]  R. Fisher.  THE STATISTICAL UTILIZATION OF MULTIPLE MEASUREMENTS , 1938 .

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

[39]  J. Bourgain.  On lipschitz embedding of finite metric spaces in Hilbert space , 1985 .

[40]  Edwin R. Hancock,et al.  Computing approximate tree edit distance using relaxation labeling , 2003, Pattern Recognit. Lett..

[41]  Nils J. Nilsson,et al.  A Formal Basis for the Heuristic Determination of Minimum Cost Paths , 1968, IEEE Trans. Syst. Sci. Cybern..

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

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

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

[45]  Horst Bunke,et al.  Bridging the Gap between Graph Edit Distance and Kernel Machines , 2007, Series in Machine Perception and Artificial Intelligence.

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