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82 Extensions to the Standard PSOM Algorithm

Here again, the similarities to Kohonen's SOM algorithm become clearly
visible: the closeness of the input x to the reference vectors w a is the im-
portant, determining factor. Here, this mechanism also tessellates the in-
put space into Voronoi cells of continuous, here parameterized, curved
maps. Similarly, the previously described smoothing schemes for discrete
cells could be applied to smooth the remaining discontinuities as well.

6.3.3 Comparison to Splines

The concept of Local-PSOM has similarities with the spline technique (Stoer
and Bulirsch 1980; Friedman 1991; Hämmerlin and Hoffmann 1991). Both
share the internal piecewise construction of a polynomial on a set of neigh-
boring nodes.

For the sub-grid size n the local PSOM becomes actually iden-

tical to the multidimensional extension of the linear splines, see Fig. 6.3d.
Each patch resembles the m-dimensional “soap-film” illustrated in Fig. 5.3
and 5.4.
An alternative was suggested by Farmer and Sidorowich (1988). The
idea is to perform a linear simplex interpolation in the jIj–dimensional
input space X in by using the jIj+1 (with respect to the input vector x)
closest support points found, which span the simplex containing Px . This
lifts the need of topological order of the training points, and the related
restriction on the number of usable points.

For n and n the local PSOM concept
resembles the quadratic
and the cubical spline, respectively. In contrast to the spline concept that
uses piecewise assembled polynomials, we employ one single, dynami-
cally constructed interpolation polynomial with the benefit of usability for
multiple dimensions m.
For m and low d the spline concept compares favorably to the poly-
nomial interpolation ansatz, since the above discussed problem of asym-
metric mapping does not occur: at each point 3 (or 4, respectively) polyno-
mials will contribute, compared with one single interpolation polynomial
in a selected node sub-grid, as described.

For m the bi-cubic, so-called tensor-product spline is usually com-
puted by row-wise spline interpolation and a column spline over the row
interpolation results (Hämmerlin and Hoffmann 1991). The procedure is
computationally very expensive and has to be independently repeated for