elmap - is a tool for fast constructing non-linear principal surfaces with
different topologies in multidimensional as well as in low-dimensional spaces,
for discrete sets of weightened points.
keywords: principal curve, principal surface, probabilistic, dimensionality reduction, nonlinear manifold, generative topographic mapping
Principal curves, graphs and surfaces are nonlinear generalizations of principal components
and subspaces, respectively. They were first defined
by Trevor Hastie and Werner Stuetzle as "self-consistent" smooth curves which
pass through the "middle" of a d-dimensional probability distribution or data
cloud. A biobliography on the subject is available in A. N. Gorban, A. Y. Zinovyev, Principal Graphs and Manifolds, In: Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods and Techniques, Olivas E.S. et al Eds. Information Science Reference, IGI Global: Hershey, PA, USA, 2009. 28–59. and
We present a novel algorithm to construct principal surfaces using methaphor
of elasticity. The picture on the right symbolizes the idea behind the
algorithm. The small points are datapoints, the big ones are a grid approximation
of a principal curve. We define an elastic energy of such system and propose
an effective algorithm to minimize it.
The methodology of principal curves and surfaces construction that we
propose has several advantages: 1) it is "natural"; 2) it is
fast; 3) it is flexible and allows many variations and adaptations; 4)
it allows easily constructing surfaces with any dimension and topology.
1) Applet for 2D data distributions (available online)
2) Contours extraction
3) Complex surface modelling
4) Multidimensional data visualization
PCA View View
on the non-linear manifold
For the moment the elastic maps algorithm has three implementations:
1) as a stand-alone full-featured tool VidaExpert fot multidimensional data visualization.
- The tool is written on Object Pascal.
2) as a C++ package for fast construction of fine grid approximations to principal surfaces.
- The package is written using standard C++, which allows to compile it on very different platforms.
It uses modestly STL. To solve quadratic optimization problem, the Sparselib++ library is used to represent
sparse matrices and IML++ library to solve system of linear equations iteratively.
3) as a part of vdaoengine Java package
- This implementation uses Colt library
for effective computations of linear systems with sparse matrices.
3) as a Java applet for 2D data distributions, available online
The sources are available upon request.
- A. Zinovyev, E. Mirkes. Data complexity measured by principal graphs, Computers and Mathemtatics with Applications (2013). http://arxiv.org/abs/1212.5841
- A. N. Gorban, A. Y. Zinovyev, Principal Graphs and Manifolds, In: Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods and Techniques, Olivas E.S. et al Eds. Information Science Reference, IGI Global: Hershey, PA, USA, 2009. 28–59.
- A. N. Gorban, A. Zinovyev, Principal manifolds and graphs in practice: from molecular biology to dynamical systems, International Journal of Neural Systems, Vol. 20, No. 3 (2010) 219–232.
- A.N. Gorban, A.Yu. Zinovyev. Elastic principal manifolds and their practical applications. Computing, 75 (4) (2005), 359 - 379
- A. Zinovyev. Method And Software For Fast Construction Of Principal Manifolds Approximations. IHES Preprint, 2003
- A.N. Gorban, A. Zinovyev. Visualization of Data by Method of Elastic Maps and Its Applications in Genomics, Economics and Sociology. IHES Preprint, 2002.
- A.N. Gorban, A. Zinovyev, D.C. Wunsch, Application of the method of elastic maps in analysis of genetic texts, in: Proceedings of International Joint Conference on Neural Networks, IJCNN-2003, Vol 3, pp. 1826–1831.
A.N. Gorban, B. Kegl, D. Wunsch, A. Zinovyev (Eds.), Principal Manifolds for Data Visualisation and Dimension Reduction, LNCSE 58, Springer: Berlin – Heidelberg – New York, 2007.
You can also have a look at the following Powerpoint presentation:
- Non-linear Principal Manifolds - elastic maps approach. (PPT)