Institute for Numerical Simulation
Rheinische Friedrich-Wilhelms-Universität Bonn

  author = {Bastian Bohn and Jochen Garcke and Michael Griebel},
  title = {A sparse grid based method for generative dimensionality
		  reduction of high-dimensional data},
  annote = {journal},
  journal = {Journal of Computational Physics},
  note = {earlier version available as INS Preprint No. 1514},
  pdf = { 1},
  doi = {10.1016/},
  volume = {309},
  number = {},
  pages = {1 - 17},
  year = {2016},
  issn = {0021-9991},
  url = {},
  keywords = {Car-crash analysis },
  abstract = {Abstract Generative dimensionality reduction methods play
		  an important role in machine learning applications because
		  they construct an explicit mapping from a low-dimensional
		  space to the high-dimensional data space. We discuss a
		  general framework to describe generative dimensionality
		  reduction methods, where the main focus lies on a
		  regularized principal manifold learning variant. Since most
		  generative dimensionality reduction algorithms exploit the
		  representer theorem for reproducing kernel Hilbert spaces,
		  their computational costs grow at least quadratically in
		  the number n of data. Instead, we introduce a grid-based
		  discretization approach which automatically scales just
		  linearly in n. To circumvent the curse of dimensionality of
		  full tensor product grids, we use the concept of sparse
		  grids. Furthermore, in real-world applications, some
		  embedding directions are usually more important than others
		  and it is reasonable to refine the underlying
		  discretization space only in these directions. To this end,
		  we employ a dimension-adaptive algorithm which is based on
		  the \{ANOVA\} (analysis of variance) decomposition of a
		  function. In particular, the reconstruction error is used
		  to measure the quality of an embedding. As an application,
		  the study of large simulation data from an engineering
		  application in the automotive industry (car crash
		  simulation) is performed. }