Explicit Magnification Control of Self-Organizing Maps for "Forbidden" Data

In this paper, we examine the scope of validity of the explicit self-organizing map (SOM) magnification control scheme of Bauer et al. (1996) on data for which the theory does not guarantee success, namely data that are n-dimensional, n > or =2, and whose components in the different dimensions are not statistically independent. The Bauer et al. algorithm is very attractive for the possibility of faithful representation of the probability density function (pdf) of a data manifold, or for discovery of rare events, among other properties. Since theoretically unsupported data of higher dimensionality and higher complexity would benefit most from the power of explicit magnification control, we conduct systematic simulations on "forbidden" data. For the unsupported n=2 cases that we investigate, the simulations show that even though the magnification exponent alpha achieved achieved by magnification control is not the same as the desired alpha desired, alpha achieved systematically follows alpha desired with a slowly increasing positive offset. We show that for simple synthetic higher dimensional data information, theoretically optimum pdf matching (alpha achieved = 1) can be achieved, and that negative magnification has the desired effect of improving the detectability of rare classes. In addition, we further study theoretically unsupported cases with real data.