Advertisement

Manifold Learning and Ranking

Chapter
  • 2.8k Downloads

Abstract

A prominent theme of this book is the spatial analysis of networks and data independent of an embedding in an ambient space. The topology and metric of the network/complex have been sufficient to define the domain upon which we may perform data analysis. However, an intrinsic metric defined on a network may be interpreted as the metric that would have been obtained if the network had been embedded into an ambient space equipped with its own metric. Consequently, it is possible to calculate an embedding map for which the induced metric approximates the intrinsic metric defined on the network. The calculation of such embeddings by manifold learning techniques is one way in which the structure of the network may be examined and visualized. A different method of examining the structure of a network is to calculate an importance ranking for each node. In contrast to the majority of this book, the ranking algorithms are generally used to examine the structure of directed graphs.

Keywords

Laplacian Matrix Ranking Algorithm Point Correspondence Locality Preserve Projection Manifold Learning 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 23.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Advances in Neural Information Processing Systems, vol. 14, pp. 585–591. MIT Press, Cambridge (2001) Google Scholar
  2. 35.
    Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge University Press, Cambridge (1994) zbMATHGoogle Scholar
  3. 60.
    Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. Computer Networks and ISDN Systems 30(1–7), 107–117 (1998) CrossRefGoogle Scholar
  4. 65.
    Burges, J.C.: Geometric methods for feature extraction and dimensional reduction. In: Data Mining and Knowledge Discovery Handbook, pp. 59–91. Springer, Berlin (2005). Chap. 1 CrossRefGoogle Scholar
  5. 71.
    Chan, T.F., Gilbert, J.R., Teng, S.H.: Geometric spectral partitioning. PARC Technical Report CSL-94-15, Xerox (1995) Google Scholar
  6. 79.
    Chung, F.: Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics 9(1), 1–19 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 86.
    Constantine, P.G., Gleich, D.F.: Using polynomial chaos to compute the influence of multiple random surfers in the PageRank model. In: Bonato, A., Chung, F.R.K. (eds.) Proc. of WAW. Lecture Notes in Computer Science, vol. 4863, pp. 82–95. Springer, Berlin (2007) Google Scholar
  8. 96.
    Dale, A.M., Fischl, B., Sereno, M.I.: Cortical surface-based analysis. I. Segmentation and surface reconstruction. NeuroImage 9(2), 179–194 (1999) CrossRefGoogle Scholar
  9. 117.
    Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. Wiley-Interscience, New York (2000) Google Scholar
  10. 130.
    Farahat, A., LoFaro, T., Miller, J., Rae, G., Ward, L.: Authority rankings from HITS, Pagerank, and SALSA: Existence, uniqueness, and effect of initialization. SIAM Journal on Scientific Computing 27(4), 1181–1201 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 136.
    Fischl, B., Sereno, M.I., Dale, A.M.: Cortical surface-based analysis. II: Inflation, flattening, and a surface-based coordinate system. NeuroImage 9(2), 195–207 (1999) CrossRefGoogle Scholar
  12. 137.
    Fischl, B., Sereno, M.I., Tootell, R.B., Dale, A.M.: High-resolution intersubject averaging and a coordinate system for the cortical surface. Human Brain Mapping 8(4), 272–284 (1999) CrossRefGoogle Scholar
  13. 140.
    Fowler, J., Jeon, S.: The authority of Supreme Court precedent. Social Networks 30(1), 16–30 (2008) CrossRefGoogle Scholar
  14. 141.
    Fowler, J., Johnson, T., Spriggs, J., Jeon, S., Wahlbeck, P.: Network analysis and the law: Measuring the legal importance of precedents at the US Supreme Court. Political Analysis 15(3), 324–346 (2007) CrossRefGoogle Scholar
  15. 149.
    Gleich, D.F.: Models and algorithms for pagerank sensitivity. Ph.D. thesis, Stanford University (2009) Google Scholar
  16. 151.
    Godsil, C., McKay, B.: Constructing cospectral graphs. Aequationes Mathematicae 25(1), 257–268 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 156.
    Gordon, C., Webb, D., Wolpert, S.: You cannot hear the shape of a drum. Bulletin of the American Mathematical Society 27, 134–138 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 193.
    He, X., Niyogi, P.: Locality preserving projections. In: Advances in Neural Information Processing Systems, vol. 16, pp. 153–160. MIT Press, Cambridge (2003) Google Scholar
  19. 203.
    Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., Stuetzle, W.: Surface reconstruction from unorganized points. In: Proc. of SIGGRAPH, vol. 26, pp. 19–26 (1992) Google Scholar
  20. 219.
    Jiang, X., Lim, L.H., Yao, Y., Ye, Y.: Statistical ranking and combinatorial Hodge theory. Mathematical Programming (to appear) Google Scholar
  21. 220.
    Kac, M.: Can one hear the shape of a drum? American Mathematical Monthly 73(4), 1–23 (1966) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 223.
    Katz, L.: A new status index derived from sociometric analysis. Psychometrika 18(1), 39–43 (1953) zbMATHCrossRefGoogle Scholar
  23. 236.
    Kleinberg, J.: Authoritative sources in a hyperlinked environment. Journal of the ACM 46(5), 604–632 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 238.
    Kohlberger, T., Uzunbas, G., Alvino, C., Kadir, T., Slosman, D., Funka-Lea, G.: Organ segmentation with level sets using local shape and appearance priors. In: Yang, G.-Z. et al. (eds.) Int. Conf. on Medical Image Comp. and Comp.-Assisted Intervention (MICCAI ’09). Lecture Notes in Computer Science, vol. 5762, pp. 34–42. Springer, Berlin (2009) CrossRefGoogle Scholar
  25. 258.
    Lévy, B.: Laplace-Beltrami eigenfunctions towards an algorithm that “understands” geometry. In: IEEE International Conference on Shape Modeling and Applications SMI 2006, p. 13 (2006) Google Scholar
  26. 274.
    Mateus, D., Horaud, R., Knossow, D., Cuzzolin, F., Boyer, E.: Articulated shape matching using Laplacian eigenfunctions and unsupervised point registration. In: IEEE Conference on Computer Vision and Pattern Recognition, 2008. CVPR 2008, pp. 1–8 (2008) Google Scholar
  27. 313.
    Qiu, H., Hancock, E.: Image segmentation using commute times. In: Proceedings of the 16th British Machine Vision Conference (BMVC 2005), pp. 929–938 (2005) Google Scholar
  28. 314.
    Qiu, H., Hancock, E.R.: Commute times, discrete Green’s functions and graph matching. In: Proc. of 13th Int. Conf. on Image Analysis and Processing—ICIAP 2005, Cagliari, Italy, September 6–8, 2005, pp. 454–462. Springer, Berlin (2005) CrossRefGoogle Scholar
  29. 318.
    Robinson, S.: The ongoing search for efficient web search algorithms. SIAM News 37(9) (2004) Google Scholar
  30. 327.
    Saerens, M., Fouss, F., Yen, L., Dupont, P.: The principal components analysis of a graph, and its relationships to spectral clustering. In: Proceedings of the 15th European Conference on Machine Learning (ECML 2004). Lecture Notes in Artificial Intelligence, pp. 371–383. Springer, Berlin (2004) CrossRefGoogle Scholar
  31. 333.
    Schoenberg, I.: Remarks to Maurice Frechet’s article “Sur la définition axiomatique d’une classe d’espace distanciés vectoriellement applicable sur l’espace de Hilbert”. Annals of Mathematics 36, 724–732 (1935) MathSciNetCrossRefGoogle Scholar
  32. 345.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), 888–905 (2000) CrossRefGoogle Scholar
  33. 365.
    Sumner, R., Popovic, J.: Deformation transfer for triangle meshes. Proceedings of SIGGRAPH 23(3), 399–405 (2004) CrossRefGoogle Scholar
  34. 373.
    Tenenbaum, J., de Silva, V., Langford, J.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000) CrossRefGoogle Scholar
  35. 387.
    Van Dam, E., Haemers, W.: Spectral characterizations of some distance-regular graphs. Journal of Algebraic Combinatorics 15(2), 189–202 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 412.
    Yen, L., Vanvyve, D., Wouters, F., Fouss, F., Verleysen, M., Saerens, M.: Clustering using a random walk based distance measure. In: Proceedings of the 13th Symposium on Artificial Neural Networks (ESANN 2005), pp. 317–324 (2005) Google Scholar
  37. 421.
    Zhang, H., van Kaick, O., Dyer, R.: Spectral mesh processing. In: Computer Graphics Forum, pp. 1–29 (2008) Google Scholar

Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  1. 1.Siemens Corporate ResearchPrincetonUSA
  2. 2.Athinoula A. Martinos Center for Biomedical Imaging, Department of RadiologyMassachusetts General Hospital, Harvard Medical SchoolCharlestownUSA

Personalised recommendations