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.