Computing

, Volume 85, Issue 4, pp 267–299

Principal manifold learning by sparse grids

Article

Abstract

In this paper, we deal with the construction of lower-dimensional manifolds from high-dimensional data which is an important task in data mining, machine learning and statistics. Here, we consider principal manifolds as the minimum of a regularized, non-linear empirical quantization error functional. For the discretization we use a sparse grid method in latent parameter space. This approach avoids, to some extent, the curse of dimension of conventional grids like in the GTM approach. The arising non-linear problem is solved by a descent method which resembles the expectation maximization algorithm. We present our sparse grid principal manifold approach, discuss its properties and report on the results of numerical experiments for one-, two- and three-dimensional model problems.

Keywords

Sparse grids Regularized principal manifolds High-dimensional data 

Mathematics Subject Classification (2000)

65N30 65F10 65N22 41A29 41A63 65D15 65D10 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institute for Numerical SimulationUniversity of BonnBonnGermany

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