About this Journal Submit a Manuscript Table of Contents
Advances in Artificial Neural Systems
Volume 2013 (2013), Article ID 486363, 8 pages
http://dx.doi.org/10.1155/2013/486363
Research Article

Visualizing Clusters in Artificial Neural Networks Using Morse Theory

Department of Mathematics, Hope College, P.O. Box 9000, Holland, MI 49422-9000, USA

Received 27 March 2013; Revised 31 May 2013; Accepted 5 June 2013

Academic Editor: Songcan Chen

Copyright © 2013 Paul T. Pearson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. Carlsson, “Topology and data,” Bulletin of the American Mathematical Society, vol. 46, no. 2, pp. 255–308, 2009. View at Publisher · View at Google Scholar · View at Scopus
  2. H. Edelsbrunner and J. Harer, Computational Topology: An Introduction, American Mathematical Society, Providence, RI, USA, 2010.
  3. A. Zomorodian, Topology for Computing, Cambridge University Press, New York, NY, USA, 2005.
  4. P. Y. Lum, G. Singh, A. Lehman et al., “Extracting insights from the shape of complex data using topology,” Scientific Reports, vol. 3, article 1236, 2013. View at Publisher · View at Google Scholar
  5. G. Singh, F. Mémoli, and G. Carlsson, “Topological methods for the analysis of high dimensional data sets and 3D object recognition,” in Eurographics Symposium on Point-Based Graphics (Prague '07), pp. 91–100.
  6. H. Adams, A. Atanasov, and G. Carlsson, “Morse theory in topological data analysis,” http://arxiv.org/abs/1112.1993.
  7. C. Marzban and U. Yurtsever, “Baby morse theory in data analysis,” in Proceedings of the Workshop on Knowledge Discovery, Modeling and Simulation (KDMS '11), pp. 15–21, August 2011. View at Publisher · View at Google Scholar · View at Scopus
  8. K.-L. Du, “Clustering: a neural network approach,” Neural Networks, vol. 23, no. 1, pp. 89–107, 2010. View at Publisher · View at Google Scholar · View at Scopus
  9. J. Herrero, A. Valencia, and J. Dopazo, “A hierarchical unsupervised growing neural network for clustering gene expression patterns,” Bioinformatics, vol. 17, no. 2, pp. 126–136, 2001. View at Scopus
  10. M. Hagan, H. Demuth, and M. Beale, Neural Network Design, PWS Publishing, Boston, Mass, USA, 1995.
  11. R. Marks and R. Reed, Neural Smithing: Supervised Learning in Feedforward Artificial Neural Networks, Denver, Bradford, UK, 1999.
  12. R. Rojas, Neural Networks: A Systematic Introduction, Springer, New York, NY, USA, 1996.
  13. G. M. Reaven and R. G. Miller, “An attempt to define the nature of chemical diabetes using a multidimensional analysis,” Diabetologia, vol. 16, no. 1, pp. 17–24, 1979. View at Scopus
  14. R. Miller, “Discussion—projection pursuit,” Annals of Statistics, vol. 13, no. 2, pp. 510–513, 1985.
  15. S. Walczak, “Methodological triangulation using neural networks for business research,” Advances in Artificial Neural Systems, vol. 2012, Article ID 517234, 12 pages, 2012. View at Publisher · View at Google Scholar
  16. M. Nicolau, A. J. Levine, and G. Carlsson, “Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival,” Proceedings of the National Academy of Sciences of the United States of America, vol. 108, no. 17, pp. 7265–7270, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. G. R. Bowman, X. Huang, Y. Yao et al., “Structural insight into RNA hairpin folding intermediates,” Journal of the American Chemical Society, vol. 130, no. 30, pp. 9676–9678, 2008. View at Publisher · View at Google Scholar · View at Scopus
  18. J. Mao and A. K. Jain, “Artificial neural networks for feature extraction and multivariate data projection,” IEEE Transactions on Neural Networks, vol. 6, no. 2, pp. 296–317, 1995. View at Publisher · View at Google Scholar · View at Scopus
  19. M. A. Kramer, “Nonlinear principal component analysis using autoassociative neural networks,” AIChE Journal, vol. 37, no. 2, pp. 233–243, 1991. View at Scopus
  20. D. De Ridder and R. P. W. Duin, “Sammon's mapping using neural networks: a comparison,” Pattern Recognition Letters, vol. 18, no. 11-13, pp. 1307–1316, 1997. View at Scopus
  21. D. K. Agrafiotis and V. S. Lobanov, “Nonlinear mapping networks,” Journal of Chemical Information and Computer Sciences, vol. 40, no. 6, pp. 1356–1362, 2000. View at Scopus
  22. W. Pedrycz, “Conditional fuzzy clustering in the design of radial basis function neural networks,” IEEE Transactions on Neural Networks, vol. 9, no. 4, pp. 601–612, 1998. View at Publisher · View at Google Scholar · View at Scopus
  23. G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Mathematics of Control, Signals, and Systems, vol. 2, no. 4, pp. 303–314, 1989. View at Publisher · View at Google Scholar · View at Scopus
  24. K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks, vol. 2, no. 5, pp. 359–366, 1989. View at Scopus
  25. P. G. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Birkhäuser, Basel, Switzerland, 2009.
  26. P. J. Huber, “Projection pursuit,” The Annals of Statistics, vol. 13, no. 2, pp. 435–475, 1985.
  27. D. F. Andrews and A. M. Herzberg, Data: A Collection of Problems from Many Fields for the Student and Research Worker, Springer, New York, NY, USA, 1985.
  28. C. R. Rao, “The use and interpretation of principal component analysis in applied research,” Sankhya Series A, vol. 26, pp. 329–358, 1964.
  29. R. J. Bolton and W. J. Krzanowski, “A characterization of principal components for projection pursuit,” The American Statistician, vol. 53, no. 2, pp. 108–109, 1999. View at Scopus
  30. C. Croux, P. Filzmoser, and M. R. Oliveira, “Algorithms for Projection-Pursuit robust principal component analysis,” Chemometrics and Intelligent Laboratory Systems, vol. 87, no. 2, pp. 218–225, 2007. View at Publisher · View at Google Scholar · View at Scopus
  31. Y. LeCun, L. Bottou, G. Orr, and K. Müller, “Efficient backprop,” in Neural Networks: Tricks of the Trade, Springer, New York, NY, USA, 1998.
  32. D. Müllner and G. Singh, “Mapper 1d for matlab,” 2013, http://comptop.stanford.edu/programs/.
  33. “Graphviz—graph visualization software,” 2013, http://www.graphviz.org/.
  34. M. Halkidi, Y. Batistakis, and M. Vazirgiannis, “On clustering validation techniques,” Journal of Intelligent Information Systems, vol. 17, no. 2-3, pp. 107–145, 2001. View at Publisher · View at Google Scholar · View at Scopus