Elastic principal manifolds and their practical applications

Gorban, A.N. and Zinovyev, A.Yu. (2004) Elastic principal manifolds and their practical applications. [Preprint]

Full text available as:



Principal manifolds defined as lines or surfaces passing through "the middle" of the data distribution serve as useful objects for many practical applications. We propose a new algorithm for fast construction of grid approximations of principal manifolds with given topology. One advantage of the method is a new form of the functional to be minimized, which becomes quadratic at the step of the vertexes positions refinement. This makes the algorithm very effective, especially for parallel implementations.

Item Type:Preprint
Keywords:principal surface, machine learning, SOM, vizualization
Subjects:Computer Science > Statistical Models
ID Code:3919
Deposited By: Gorban, Prof Alexander N.
Deposited On:06 Nov 2004
Last Modified:11 Mar 2011 08:55

References in Article

Select the SEEK icon to attempt to find the referenced article. If it does not appear to be in cogprints you will be forwarded to the paracite service. Poorly formated references will probably not work.

Aizenberg, L. Carleman's Formulas in Complex Analysis: theory and Applications. Mathematics and Its Applications; 244. Kluwer, 1993.

Banfield J. D., Raftery A.E. Ice floe identification in satellite images using mathematical morphology and clustering about principal curves. Journal of the American Statistical Association 87, N 417, pp. 7-16, 1992.

Cai W., Shao X., Maigret B. Protein-ligand recognition using spherical harmonic molecular surfaces: towards a fast and efficient filter for large virtual throughput screening. J. Mol. Graph. Model. 2002 Jan; 20(4): 313-28.

Dongarra J., Lumsdaine A., Pozo R., Remington K. A Sparse Matrix Library in C++ for High Performance Architectures. Proceedings of the Second Object Oriented Numerics Conference, pp. 214-218, 1994.

Erwin E., Obermayer K., Schulten K. Self-organizing maps: ordering, convergence properties and energy functions. Biological Cybernetics, 67:47–55, 1992.

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

Gorban A.N., Pitenko A.A., Zinov'ev A.Y., Wunsch D.C. Vizualization of any data using elastic map method. 2001. Smart Engineering System Design 11, p. 363-368.

Gorban A.N., Rossiev A., Makarenko N., Kuandykov Y., Dergachev V. Recovering data gaps through neural network methods. International Journal of Geomagnetism and Aeronomy, 2002, V 3, N. 2.

Gorban A.N., Zinovyev A. Yu. Method of Elastic Maps and its Applications in Data Visualization and Data Modeling. International Journal of Computing Anticipatory Systems,

CHAOS. 2001. V 12. PP. 353-369.

Gorban A.N., Zinovyev A., Wunsch D.C. Application of the method of elastic maps in analysis of genetic texts. In Proceedings of IJCNN2003. 2003.

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. 2001. M/01/36.

Gorban, A.N., Rossiev, A.A. Neural network iterative method of principal curves for data with gaps. Journal of Computer and System Sciences International 38(5), pp. 825-831, 1999.

Gorban A.N., Zinovyev A.Yu. Pitenko A.A.Visualization of data. Method of elastic maps (in Russian). Neurocomputers, 2002. N4. p.19-30.

Gorban A.N., Zinovyev A.Yu., Pitenko A.A. Visualization of data using method of elastic maps (in Russian). Informatsionnie technologii. 'Mashinostrornie' Publ., Moscow, 2000. N6, P.26-35.

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

Hastie T., Stuetzle W. Principal curves. Journal of the American Statistical Association 84, N 406. pp. 502-516, 1989.

Xie H., Qin H. A Physics-based Framework for Subdivision Surface Design with Automatic

Rules Control. In Proceedings of the Tenth Pacific Conference on Computer Graphics and

Applications (Pacific Graphics 2002), IEEE Press, pp. 304-315.

Kohonen, T. (1982) Self-organized formation of topologically correct feature maps. Biological Cybernetics, 43:59-69.

Kegl B. Principal curves: learning, design, and applications, Ph. D. Thesis, Concordia University, Canada, 1999.

Kegl B., Krzyzak A. Piecewise linear skeletonization using principal curves. IEEE Transactions on Pattern Analysis and Machine Intelligence 24, N 1, pp. 59-74, 2002.

Kegl B., Krzyzak A., Linder T., Zeger K. A polygonal line algorithm for constructing principal curves. Neural Information Processing Systems 1998. MIT Press, 1999, pp. 501-507.

Kegl B., Krzyzak A., Linder T., Zeger K. Learning and design of principal curves. IEEE Transactions on Pattern Analysis and Machine Intelligence 22, N 3, pp. 281-297, 2000.

LeBlanc M., Tibshirani R. Adaptive principal surfaces. Journal of the American Statistical Association 89, pp. 53-64, 1994.

Mandal C., Qin H., Vemuri B.C. A novel FEM-based dynamic framework for subdivision surfaces. Comp.-Aided Design 32 (2000), pp. 479-497.

Mulier F., Cherkassky V. Self-organization as an iterative kernel smoothing process. Neural Computation, 7:1165–1177, 1995.

Ritter H., Martinetz T., Schulten K.. Neural Computation and Self-Organizing Maps: An Introduction. Addison-Wesley, Reading, Massachusetts, 1992.

Roweis S. and Saul L. K. Nonlinear dimensionality reduction by locally linear embedding. 2000. Science, V 290, pp. 2323-2326.

Sayle R., Bissell A. RasMol: A Program for Fast Realistic Rendering of Molecular Structures with Shadows. 1992. In Proceedings of the 10th Eurographics UK'92 Conference, University of Edinburgh, Scotland.

Stanford D., Raftery A.E. Principal curve clustering with noise. IEEE Transactions on Pattern Analysis and Machine Intelligence 22, N 6, pp.601-609, 2000.

Tenenbaum J. B., De Silva V. and Langford J. C. A global geometric framework for nonlinear dimensionality reduction. 2000. Science 290, pp. 2319-2323.

Verbeek J.J., Vlassis N., Krose B. A k-segments algorithm for finding principal curves. A technical report.

Zinovyev A.Yu., Pitenko A.A. Popova T.G. Practical applications of the method of elastic maps (in Russian). Neurocomputers, 2002. N4. p.31-39.

Zinovyev A. Visualization of Multidimensional Data. Russia, Krasnoyarsk State University Press Publ., 2000, 168 p


Repository Staff Only: item control page