Title:  "Nonlinear Dimensionality Reduction"

                   Jose Costa
   Center for the Mathematics of Information
       California Institute of Technology

How can one make sense of the enormous quantity of information
available in current and future databases and use it in a meaningful
way, from a simple problem of visualizing a relevant data
characteristic to more complex decision making tasks?

In this second lecture on high-dimensional data processing, we will
discuss the problem of dimensionality reduction, i.e., finding lower
dimensional representations for high-dimensional data with little or
no loss of content information. By assuming that the data sets lie on
a low dimensional manifold, one can design algorithms that overcome
the limitations of classical linear methods such as Principal
Component Analysis or Multidimensional Scaling.  In this lecture, we
will give an overview of the many flavors proposed to the address the
manifold learning problem as well as discuss its major fault. Although
dimensionality reduction is usually invoked as a tool to improve
classification, regression, denoising or visualization tasks, current
algorithms do not use this information to find a particular lower
dimensional representation of the data. We will show how to
incorporate this information in the context of supervised and
semi-supervised learning.