Some steps towards a general principle for dimensionality reduction mappings

Nov, 2010

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

Link

@INPROCEEDINGS{Hammer-DROPS-2010,
author = {Barbara Hammer and Kerstin Bunte and Michael Biehl},
title = {Some steps towards a general principle for dimensionality reduction mappings},
editor = {B. Hammer and P. Hitzler and W. Maas and M. Toussaint},
booktitle = {Learning paradigms in dynamic environments},
year = {2010},
publisher = {Schloss Dagstuhl, Leibniz Zentrum f\"ur Informatik},
series = {Dagstuhl Seminar Proceedings},
volume = {10302},
pages = {15},
month = {"Nov."},
url = {http://drops.dagstuhl.de/opus/volltexte/2010/2803/pdf/10302.HammerBarbara.Paper.2803.pdf},
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},
}