Parameterizing high-dimensional data sets with kernel map manifolds

Many important data analysis problems come in in the form of a set of data points each of which contains a large number of measurements, which can be considered scattered data in a very high dimensional space. Visualizing and analyzing such data is challenging, because the dimensionality of the ambient space makes visualization and statistical analysis quite difficult. However, often such data sets do not fill the ambient space, but rather lie close to some lower dimensional manifold. If the manifold is linear, then principal component analysis and other linear models can extract the best fitting models. However, the nonlinear case demands a more sophisticated set of tools for learning the underlying structure of high-dimensional data. This talk examines the problem of manifold learning from a machine learning point of view and describes new tools that make the connection between manifold learning and the statistical generalization of PCA, called principal surfaces. We also present results on examples of visualization and analysis of high-dimensional data from graphics, perception, and medicine.