Principal manifold learning by sparse grids

Authors:
Christian Feuersänger;Michael Griebel
Affiliations:
University of Bonn, Institute for Numerical Simulation, Bonn, Germany;University of Bonn, Institute for Numerical Simulation, Bonn, Germany
Venue:
Computing
Year:
2009

Citing 20
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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.