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2.5. NON-LINEAR 29
technique but rather an algorithm for feature selection, which we will briefly
comment on in the end of this chapter.
In the next section we will go through a set of algorithms that uses local
similarity measures in the data to find kernels onto which a spectral decom-
position can be applied to find a geometrical representation of the data.
2.5.2 Proximity Graph Methods
Several dimensionality reduction algorithms have been suggested that are
based on local similarity measures in the data. These algorithms are based
on a proximity graph [66, 29, 10] extracted from the data. A proximity graph
is a graph that represent a specific neighborhood relationships in the data.
In the graph each node corresponds to a data point, vertices connects nodes
that are related through the specified relationship potentially associated with
a vertex weight. The fundamental idea behind proximity graph based algo-
rithms for dimensionality reduction is that locally the data can be assumed to
lie on a linear manifold. This means that locally the distance in the original
representation of the data will be a good approximation to the manifold dis-
tance. Therefore the neighborhood relationship used for proximity graphs in
dimensionality reduction is the inter-distance between points in the original
representation. Usually the graphs are constructed either from an N near-
est neighbor algorithm where the N closest points are connected or from a
ǫ nearest neighbor where all points within a ball of radius ǫ are connected.
Setting either parameter is of significant importance as only points whose
inter-distance can be assumed to approximate the manifold distance should