Project Details
Projekt
Lower-dimensional principal manifold learning in higher-dimensional data spaces by sparse grid methods
Applicant
Professor Dr. Michael Griebel
Subject Area
Mathematics
Term
from 2008 to 2013
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 79706051
In many areas of science, it can be observed that problems of high dimensions possess only low effective dimensionality. This may be exploited to circumvent the curse of dimensionality of a numerical approximation. The main problem, however, is to find a proper coordinate system and the best associated effective dimension. This is especially difficult if one allows nonlinear intrinsic coordinates which however often allow for a substantially better dimension reduction than a linear approach by the PCA. In this project, we use the approach of regularized principal manifolds to compute such coordinates. Since the remaining dimensions may nevertheless be more than three, we employ sparse grids for the representation of the coordinate functions of the manifold. This way, the numerical complexity is further reduced and problems with up to 20 intrinsic dimensions may be handled efficiently. We will derive the effective dimension and the intrinsic coordinates automatically by a dimension-adaptive sparse grid method using dimension sensitive error indicators.
DFG Programme
Priority Programmes
