A problem with the nonlinear MDS methods is that they are
computationally very intensive for large data sets. The computational
complexity can be reduced, however, by restricting attention to a
subset of the distances between the data items. When placing a point
on a plane its distance from two other points of the plane can be set
exactly. This property is used in the *triangulation method*
[Lee et al., 1977]. Points are mapped sequentially onto the plane, and the
distance of the new item to the two nearest items already mapped is
preserved. Alternatively, the distance to the nearest item and a
reference point that is common to all items may be preserved. The
points are mapped in such an order that all of the nearest-neighbor
distances in the original space will be preserved. The triangulation
can be computed quickly, compared to the MDS methods, but since it
only tries to preserve a small fraction of the distances the
projection may be difficult to interpret for large data sets. The
method may, however, be useful in connection with Sammon's mapping
[Biswas et al., 1981].

The dimensionality of data sets can also be reduced with the aid of
autoassociative neural networks that represent their inputs using a
smaller number of variables than there are dimensions in the input
data. Such networks try to reconstruct their inputs as faithfully as
possible, and the representation of the data items constructed into
the network can be used as the reduced-dimensional expression of the
data. Some linear and nonlinear associative memories have been
introduced by Kohonen (1984). The
representations formed into the hidden layer of a multilayer
perceptron have also been used for the dimension reduction task
[DeMers and Cottrell, 1993, Garrido et al., 1995]. A special version of the multilayer
perceptrons, a *replicator neural network* [Hecht-Nielsen, 1995]
has even been shown *capable* of representing its inputs in terms
of their ``natural coordinates''. This occurs for a somewhat idealized
model when the inherent dimensionality *q* of the data increases. The
natural coordinates correspond to coordinates in a *q*-dimensional
unit cube that has been transformed elastically to fit the
distribution of the data. The inherent dimensionality of
the data is, of course, difficult to identify in practice.

The replicator neural networks could possibly be used for forming a
visualization of the data set by choosing *q*=2. Although intriguing,
the approach would require a separate study that would compare both
the quality of the results and the computational requirements for a
network having a practical size. The learning of the multilayer
perceptrons with the backpropagation algorithm (cf., e.g., Rumelhart
et al., 1986) is known to be very slow
[Haykin, 1994], but it is possible that some alternative learning
algorithms would be more feasible.

Mon Mar 31 23:43:35 EET DST 1997