Abstract
The aim in this paper is to show how the problem of learning the modes of structural variation in sets of graphs can be solved by converting the graphs to strings. We commence by showing how the problem of converting graphs to strings, or seriation, can be solved using semi-definite programming (SDP). This is a convex optimisation procedure that has recently found widespread use in computer vision for problems including image segmentation and relaxation labelling. We detail the representation needed to cast the graph-seriation problem in a matrix setting so that it can be solved using SDP. We show how to perform PCA on the strings delivered by our method. By projecting the seriated graphs on to the leading eigenvectors of the sample covariance matrix, we pattern spaces suitable for graph clustering.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bagdanov, A.D., Worring, M.: First order Gaussian graphs for efficient structure classification. Pattern Recognition 36(6), 1311–1324 (2003)
Wong, A.K.C., Constant, J., You, M.: Random graphs. Syntactic and Structural Pattern Recognition-Fundamentals, Advances, and Applications (1990)
Robles-Kelly, A., Hancock, E.R.: Graph Edit Distance from Spectral Seriation. IEEE Transactions on Pattern Analysis and Machine Intelligence (2004) (To appear)
Harris, C., Stephens, M.: A combined corner and edge detector. In: Fourth Alvey Vision Conference, pp. 147–151 (1988)
Christmas, W.J., Kittler, J., Petrou, M.: Structural matching in computer vision using probabilistic relaxation. IEEE Transactions on Pattern Analysis and Machine Intelligence 17(8), 749–764 (1995)
Schellewald, C., Schnőrr, C.: Subgraph Matching with Semidefinite Programming. In: Proceedings IWCIA (International Workshop on Combinatorial Image Analysis, Palermo, Italy (2003)
Wolkowicz, H., Zhao, Q.: Semidefinite Programming relaxation for the graph partitioning problem. Discrete Appl. Math., 461–479 (1999)
Keuchel, J., Schnörr, C., Schellewald, C., Cremers, D.: Binary Partitioning, Perceptual Grouping, and Restoration with Semidefinite Programming. IEEE Trans. Pattern Analysis and Machine Intelligence 25(11), 1364–1379 (2003)
Atkins, J.E., Boman, E.G., Hendrickson, B.: A Spectral Algorithm for Seriation and the Consecutive Ones Problem. SIAM Journal on Computing 28(1), 297–310 (1998)
Fujisawa, K., Futakata, Y., Kojima, M., Nakata, K., Yamashita, M.: Sdpa-m user’s manual, http://sdpa.is.titech.ac.jp/SDPA-M
Vandenberghe, L., Boyd, S.: Semidefinite Programming. SIAM Review 38(1), 49–95 (1996)
Veldhorst, M.: Approximation of the consecutive one matrix augmentation problem. J. Comput. 14, 709–729 (1985)
Goemans, M.X., WIlliamson, D.P.: Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming. J. ACM 42(6), 1115–1145 (1995)
Wilson, R.C., Hancock, E.R.: Structural Matching by Discrete Relaxation. IEEE Transactions on Pattern Analysis and Machine Intelligence 19(6), 634–648 (1997)
Rao, S., Richa, A.W.: New approximation techniques for some ordering problems. ACM-SIAM Symposium on Discrete Algorithms, 211–218 (1998)
Boykov, Y., Veksler, O., Zabih, R.: Fast Approximate Energy Minimization via Graph Cuts. IEEE transactions on Pattern Analysis and Machine Intelligence 23(11), 1222–1239 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yu, H., Hancock, E.R. (2005). Eigenspaces from Seriated Graphs. In: Gagalowicz, A., Philips, W. (eds) Computer Analysis of Images and Patterns. CAIP 2005. Lecture Notes in Computer Science, vol 3691. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11556121_23
Download citation
DOI: https://doi.org/10.1007/11556121_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28969-2
Online ISBN: 978-3-540-32011-1
eBook Packages: Computer ScienceComputer Science (R0)