Skip to main content
SpringerLink
Log in
Book cover

Modeling, Simulation and Optimization of Complex Processes - HPSC 2012 pp 51–62Cite as

A Sparse Grid Based Generative Topographic Mapping for the Dimensionality Reduction of High-Dimensional Data

  • Conference paper
  • First Online:

  • 785 Accesses

Abstract

Most high-dimensional data exhibit some correlation such that data points are not distributed uniformly in the data space but lie approximately on a lower-dimensional manifold. A major problem in many data-mining applications is the detection of such a manifold from given data, if present at all. The generative topographic mapping (GTM) finds a lower-dimensional parameterization for the data and thus allows for nonlinear dimensionality reduction. We will show how a discretization based on sparse grids can be employed for the mapping between latent space and data space. This leads to efficient computations and avoids the ‘curse of dimensionality’ of the embedding dimension. We will use our modified, sparse grid based GTM for problems from dimensionality reduction and data classification.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    We can replace | l | 1 by \(\vert \mathbf{l}\vert _{1} + \vert \{s: l_{s} = 0\}\vert \) in (13), which leads to a slightly different treatment of boundary functions, but has otherwise the same asymptotic properties, see [9].

References

  1. Bache, K., Lichman, M.: UCI Machine Learning Repository. http://archive.ics.uci.edu/ml (2012)

  2. Balder, R., Zenger, C.: The solution of multidimensional real Helmholtz equations on sparse grids. SIAM J. Sci. Comput. 17, 631–646 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bishop, C., James, G.: Analysis of multiphase flows using dual-energy gamma densitometry and neural networks. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 327(2–3), 580–593 (1993)

    Article  Google Scholar 

  4. Bishop, C., Svensen, M., Williams, C.: GTM: the generative topographic mapping. Neural Comput. 10(1), 215–234 (1998)

    Article  Google Scholar 

  5. Bungartz, H.: Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. Dissertation, Fakultät für Informatik, Technische Universität München (1992)

    Google Scholar 

  6. Bungartz, H., Griebel, M.: Sparse grids. Acta Numer. 13, 1–123 (2004)

    Article  MathSciNet  Google Scholar 

  7. Craven, P., Wahba, G.: Smoothing noisy data with spline functions. Numer. Math. 31(4), 377–403 (1978)

    Article  MathSciNet  Google Scholar 

  8. Dempster, A., Laird, N., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. 39, 1–38 (1977)

    MathSciNet  MATH  Google Scholar 

  9. Feuersänger, C.: Sparse Grid Methods for Higher Dimensional Approximation. Südwest-deutscher Verlag für Hochschulschriften AG & Company KG, Saarbrücken (2010)

    Google Scholar 

  10. Feuersänger, C., Griebel, M.: Principal manifold learning by sparse grids. Computing 85(4), 267–299 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gerstner, T., Griebel, M.: Dimension–adaptive tensor–product quadrature. Computing, 71(1), 65–87 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gorman, R., Sejnowski, T.: Analysis of hidden units in a layered network trained to classify sonar targets. Neural Netw. 1, 75 (1988)

    Article  Google Scholar 

  13. Griebel, M., Hullmann, A.: Dimensionality reduction of high-dimensional data with a nonlinear principal component aligned generative topographic mapping. SIAM J. Sci. Comput. 36(3), A1027–A1047 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hullmann, A.: Schnelle varianten des generative topographic mapping. Diploma thesis, Institute for Numerical Simulation, University of Bonn (2009)

    Google Scholar 

  15. Kohavi, R.: A study of cross-validation and bootstrap for accuracy estimation and model selection. In: Proceedings of the 14th International Joint Conference on Artificial Intelligence – Volume 2 (IJCAI’95), San Francisco, pp. 1137–1143. Morgan Kaufmann (1995)

    Google Scholar 

  16. Kullback, S.: Information Theory and Statistics. Wiley, New York (1959)

    MATH  Google Scholar 

  17. Lee, J., Verleysen, M.: Nonlinear Dimensionality Reduction. Springer, New York/London (2007)

    Book  MATH  Google Scholar 

  18. Neal, R., Hinton, G.: A view of the EM algorithm that justifies incremental, sparse, and other variants. In: Learning in Graphical Models, pp. 355–368. Kluwer Academic, Dordrecht/Boston (1998)

    Google Scholar 

  19. Pflüger, D., Peherstorfer, B., Bungartz, H.: Spatially adaptive sparse grids for high-dimensional data-driven problems. J. Complex. 26(5), 508–522 (2010)

    Article  MATH  Google Scholar 

  20. Schölkopf, B., Smola, A.: Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT, Cambridge (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Numerical Simulation, University of Bonn, Bonn, Germany

    Michael Griebel & Alexander Hullmann

Authors
  1. Michael Griebel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Hullmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Hullmann .

Editor information

Editors and Affiliations

  1. Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Heidelberg, Germany

    Hans Georg Bock

  2. Vietnam Academy of Science and Technology(VAST), Hanoi, Vietnam

    Xuan Phu Hoang

  3. Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Heidelberg, Germany

    Rolf Rannacher

  4. Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Heidelberg, Germany

    Johannes P. Schlöder

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Griebel, M., Hullmann, A. (2014). A Sparse Grid Based Generative Topographic Mapping for the Dimensionality Reduction of High-Dimensional Data. In: Bock, H., Hoang, X., Rannacher, R., Schlöder, J. (eds) Modeling, Simulation and Optimization of Complex Processes - HPSC 2012. Springer, Cham. https://doi.org/10.1007/978-3-319-09063-4_5

Download citation

Publish with us

Policies and ethics

Search

Navigation