Skip to main content

Elastic Principal Graphs and Manifolds and their Practical Applications


Principal manifolds serve as useful tool for many practical applications. These manifolds are defined as lines or surfaces passing through “the middle” of data distribution. We propose an algorithm for fast construction of grid approximations of principal manifolds with given topology. It is based on analogy of principal manifold and elastic membrane. First advantage of this method is a form of the functional to be minimized which becomes quadratic at the step of the vertices position refinement. This makes the algorithm very effective, especially for parallel implementations. Another advantage is that the same algorithmic kernel is applied to construct principal manifolds of different dimensions and topologies. We demonstrate how flexibility of the approach allows numerous adaptive strategies like principal graph constructing, etc. The algorithm is implemented as a C++ package elmap and as a part of stand-alone data visualization tool VidaExpert, available on the web. We describe the approach and provide several examples of its application with speed performance characteristics.

This is a preview of subscription content, access via your institution.


  • Aizenberg, L.: Carleman’s formulas in complex analysis: theory and applications. Math. Appl., 244. Kluwer 1993.

  • J. D. Banfield A. E. Raftery (1992) ArticleTitleIce flow identification in satellite images using mathematical morphology and clustering about principal curves J. Am. Stat. Assoc. 87 IssueID417 7–16

    Google Scholar 

  • M. Born K. Huang (1954) Dynamical theory of crystal lattices Oxford University Press Oxford

    Google Scholar 

  • W. Cai X. Shao B. Maigret (2002) ArticleTitleProtein-ligand recognition using spherical harmonic molecular surfaces: towards a fast and efficient filter for large virtual throughput screening. J. Mol. Graph. Model 20 IssueID4 313–28 Occurrence Handle10.1016/S1093-3263(01)00134-6 Occurrence Handle11858640

    Article  PubMed  Google Scholar 

  • V. A. Dergachev A. N. Gorban A. A. Rossiev L. M. Karimova E. B. Kuandykov N. G. Makarenko P. Steier (2001) ArticleTitleThe filling of gaps in geophysical time series by artificial neural networks Radiocarbon 43 IssueID2A 365–371

    Google Scholar 

  • Dongarra, J., Lumsdaine, A., Pozo, R., Remington, K.: A sparse matrix library in C++ for high performance architectures. In: Proc. 2nd Object Oriented Numerics Conference, pp. 214–218, 1994.

  • Elmap: C++ package available online: zinovyev/vidaexpert/elmap.

  • E. Erwin K. Obermayer K. Schulten (1992) ArticleTitleSelf-organizing maps: ordering, convergence properties and energy functions Biol. Cybern. 67 47–55 Occurrence Handle1606243

    PubMed  Google Scholar 

  • Gorban, A. N. (ed.): Methods of neuroinformatics (in Russian). Krasnoyarsk State University Press, p. 205, 1998.

  • Gorban, A. N., Karlin, I. V., Zinovyev, A. Yu.: Invariant grids for reaction kinetics. Physica A 333, 106–154 (2004). Preprint online:–42.html.

    Google Scholar 

  • Gorban, A. N., Karlin, I. V., Zinovyev, A. Yu.: Constructive methods of invariant manifolds for kinetic problems. Phys. Reports 396(4–6), 197–403 (2004). Preprint online:

    Google Scholar 

  • A. N. Gorban A. A. Pitenko A. Y. Zinov’ev D. C. Wunsch (2001) ArticleTitleVizualization of any data using elastic map method Smart Eng. Syst. Des. 11 363–368

    Google Scholar 

  • A. N. Gorban A. A. Rossiev (1999) ArticleTitleNeural network iterative method of principal curves for data with gaps J. Comput. Sys. Sci. Int. 38 IssueID5 825–831

    Google Scholar 

  • A. Gorban A. Rossiev N. Makarenko Y. Kuandykov V. Dergachev (2002) ArticleTitleRecovering data gaps through neural network methods Int. J. Geomagnetism Aeronomy 3 IssueID2 191–197

    Google Scholar 

  • Gorban, A. N., Rossiev, A. A., Wunsch, D. C. II: Neural network modeling of data with gaps: Method of principal curves, Carleman’s formula, and other. In: USA–NIS Neurocomputing Opportunities Workshop, Washington, July 1999 (Associated with IJCNN’99). Preprint online:

  • Gorban, A. N., Zinovyev, A. Yu.: Visualization of data by method of elastic maps and its applications in genomics, economics and sociology. Preprint of Institut des Hautes Etudes Scientiques, M/01/36, 2001.

  • A. N. Gorban A. Yu. Zinovyev (2001) ArticleTitleMethod of elastic maps and its applications in data visualization and data modeling Int. J. Comput. Anticipatory Syst. CHAOS 12 353–369

    Google Scholar 

  • A. N. Gorban A. Yu. Zinovyev A. A. Pitenko (2000) ArticleTitleVisualization of data using method of elastic maps (in Russian) Informatsionnie Technologii 6 26–35

    Google Scholar 

  • A. N. Gorban A. Yu. Zinovyev A. A. Pitenko (2002) ArticleTitleVisualization of data. Method of elastic map (in Russian) Neurocomputers 4 19–30

    Google Scholar 

  • Gorban, A. N., Zinovyev, A. Yu., Wunsch, D. C.: Application of the method of elastic maps in analysis of genetic texts. In: Proc. Int. Joint Conf. on Neural Networks (IJCNN), Portland, July 20–24, 2003.

  • A. Gusev (2004) ArticleTitleFinite element mapping for spring network representations of the mechanics of solids Phys. Rev. Lett. 93 IssueID2 034302 Occurrence Handle10.1103/PhysRevLett.93.034302 Occurrence Handle15323824

    Article  PubMed  Google Scholar 

  • Hastie, T.: Principal curves and surfaces. PhD Thesis, Stanford University, 1984.

  • T. Hastie W. Stuetzle (1989) ArticleTitlePrincipal curves J. Am. Stat. Assoc. 84 IssueID406 502–516

    Google Scholar 

  • T. Kohonen (1982) ArticleTitleSelf-organized formation of topologically correct feature maps Biol. Cybern. 43 59–69 Occurrence Handle10.1007/BF00337288

    Article  Google Scholar 

  • Kégl, B.: Principal curves: learning, design, and applications. PhD Thesis, Concordia University, Canada, 1999.

  • B. Kégl A. Krzyzak (2002) ArticleTitlePiecewise linear skeletonization using principal curves IEEE Trans. Pattern Anal. Machine Intell. 24 IssueID1 59–74 Occurrence Handle10.1109/34.982884

    Article  Google Scholar 

  • Kégl, B., Krzyzak, A., Linder, T., Zeger, K.: A polygonal line algorithm for constructing principal curves. Neural Inf. Processing Sys. pp. 501–507 (1999).

  • B. Kégl A. Krzyzak T. Linder K. Zeger (2000) ArticleTitleLearning and design of principal curves IEEE Trans. Pattern Anal. Machine Intell. 22 IssueID2 281–297 Occurrence Handle10.1109/34.841759

    Article  Google Scholar 

  • M. LeBlanc R. Tibshirani (1994) ArticleTitleAdaptive principal surfaces J. Amer. Stat. Assoc. 89 53–64

    Google Scholar 

  • C. Mandal H. Qin B. C. Vemuri (2000) ArticleTitleA novel FEM-based dynamic framework for subdivision surfaces Comp. Aided Des. 32 479–497 Occurrence Handle10.1016/S0010-4485(00)00037-3

    Article  Google Scholar 

  • F. Mulier V. Cherkassky (1995) ArticleTitleSelf-organization as an iterative kernel smoothing process Neural Comput. 7 1165–1177 Occurrence Handle7584895

    PubMed  Google Scholar 

  • Ritter, H., Martinetz, T., Schulten, K.: Neural computation and self-organizing maps: an introduction. Addison-Wesley Reading, Massa. 1992.

  • S. Roweis L. K. Saul (2000) ArticleTitleNonlinear dimensionality reduction by locally linear embedding Science 290 2323–2326 Occurrence Handle10.1126/science.290.5500.2323 Occurrence Handle11125150

    Article  PubMed  Google Scholar 

  • Sayle, R., Bissell, A.: RasMol: a program for fast realistic rendering of molecular structures with shadows. In: Proc. 10th Eurographics UK’92 Conf., University of Edinburgh, Scotland, 1992.

  • D. Stanford A. E. Raftery (2000) ArticleTitlePrincipal curve clustering with noise IEEE Trans. Pattern Anal. Machine Intell. 22 IssueID6 601–609 Occurrence Handle10.1109/34.862198

    Article  Google Scholar 

  • J. B. Tenenbaum V. DeSilva J. C. Langford (2000) ArticleTitleA global geometric framework for nonlinear dimensionality reduction Science 290 2319–2323 Occurrence Handle10.1126/science.290.5500.2319 Occurrence Handle11125149

    Article  PubMed  Google Scholar 

  • Van Gelder, A., Wilhelms, J.: Simulation of elastic membranes and soft tissue with triangulated spring meshes. Technical Report: UCSC-CRL-97–12, 1997.

  • Verbeek, J. J., Vlassis, N., Krose, B.: A k-segments algorithm for finding principal curves. Technical report, 2000. Online:

  • VidaExpert: Stand-alone application for multidimensional data visualization. Available online: zinovyev/vidaexpert/vidaexpert.htm.

  • Xie, H., Qin, H.: A physics-based framework for subdivision surface design with automatic rules control. In: Proc. 10th Pacific Conf. on Computer Graphics and Applications (Pacific Graphics 2002), IEEE Press, 304–315, 2002.

  • Zinovyev, A.: Visualization of multidimensional data. Krasnoyarsk State University Press Publ., 2000.

  • A. Yu. Zinovyev A. N. Gorban T. G. Popova (2003) ArticleTitleSelf-organizing approach for automated gene identification Open Sys. Inf. Dyn. 10 IssueID4 321–333 Occurrence Handle10.1023/B:OPSY.0000009554.93005.f6 Occurrence HandleMR2026858

    Article  MathSciNet  Google Scholar 

  • A. Yu. Zinovyev A. A. Pitenko T. G. Popova (2002) ArticleTitlePractical applications of the method of elastic maps (in Russian) Neurocomputers 4 31–39

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to A. Gorban.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gorban, A., Zinovyev, A. Elastic Principal Graphs and Manifolds and their Practical Applications. Computing 75, 359–379 (2005).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:

AMS Subject Classifications