Forbidden magnification? II

The twin of this paper, “Forbidden Magnification? I.” [1], presents systematic SOM simulations with the explicit magnification control scheme of Bauer, Der, and Herrmann [2] on data for which the theory does not guarantee success, namely data that are n-D, n > 2 and/or data whose components in the different dimensions are not statistically independent. For the unsupported n = 2 cases that we investigated the simulations show that even though the magnification exponent αachieved achieved by magnification control is not the same as the intended αintended, the direction and sign of αachieved systematically follows αintended with a more or less constant offset. We experimentally showed that for simple synthetic higher dimensional data negative magnification has the desired effect of improving the detectability of rare classes. In this paper we study further theoretically unsupported cases, including experiments with real data. 1 Known limits of SOM magnification control Controlling the magnification of Self-Organizing Neural Maps (i.e. the functional relationship between the pdf of the input data and the density of the SOM weights in the input space) is an extremely attractive possibility because various values of the magnification exponent, denoted by α in this paper, effect desirable quantization properties. For example, α = 1 realizes maximum entropy quantizaton, α = 1/3 and α = 1/2 force minimum distortion quantization for 1and 2-D data, respectively. As is known, the basic Kohonen SOM’s inherent property is a map magnification of α = 2/3. The twin of this paper, [1], and references therein give more details on magnification. The algorithm by Bauer, Der, and Herrmann [2] (referred to as BDH from now on) provided a principled approach to obtaining a desired magnification exponent for 1-D data and for 2-D data whose components are statistically independent. Most real data, of course, do not obey the above conditions, yet it is real data scenarios that would benefit the most from explicit magnification control. After careful verification of known magnification properties on “allowed” data we examined Request color reprint by email. ∗Authors are partially supported by the Applied Information Systems Research Program of NASA, Office of Space Science, NAG9-10432. ESANN'2004 proceedings European Symposium on Artificial Neural Networks Bruges (Belgium), 28-30 April 2004, d-side publi., ISBN 2-930307-04-8, pp. 57-62