Some steps towards a general principle for dimensionality reduction mappings

Abstract

In the past years, many dimensionality reduction methods have been
established which allow to visualize high dimensional data sets. Recently,
also formal evaluation schemes have been proposed for data visualization,
which allow a quantitative evaluation along general principles. Most techniques
provide a mapping of a priorly given finite set of points only, requiring
additional steps for out-of-sample extensions. We propose a general
view on dimensionality reduction based on the concept of cost functions,
and, based on this general principle, extend dimensionality reduction to
explicit mappings of the data manifold. This offers the possibility of simple
out-of-sample extensions. Further, it opens a way towards a theory
of data visualization taking the perspective of its generalization ability
to new data points. We demonstrate the approach based in a simple
example.