Magnification factors specify the extent to which the area of a small patch of the latent (or `feature') space of a topographic mapping is magnified on projection to the data space, and are of considerable interest in both neuro-biological and data analysis contexts. Previous attempts to consider magnification factors for the self-organizing map (SOM) algorithm have been hindered because the mapping is only defined at discrete points (given by the reference vectors). In this paper we consider the batch version of SOM, for which a continuous mapping can be defined, as well as the Generative Topographic Mapping (GTM) algorithm of Bishop et al. (1997) which has been introduced as a probabilistic formulation of the SOM. We show how the techniques of differential geometry can be used to determine magnification factors as continuous functions of the latent space coordinates. The results are illustrated here using a problem involving the identification of crab species from morphological data.
|Publication status||Published - 1997|
|Event||Proceedings 1997 Workshop on Self-Organizing Maps - Helsinki University of Technology, Finland|
Duration: 4 Jun 1997 → 6 Jun 1997
|Workshop||Proceedings 1997 Workshop on Self-Organizing Maps|
|City||Helsinki University of Technology|
|Period||4/06/97 → 6/06/97|
- Magnification factors
- topographic mapping
- Generative Topographic Mapping
- differential geometry
Bishop, C. M., Svens'en, M., & Williams, C. K. I. (1997). Magnification factors for the SOM and GTM algorithms. Paper presented at Proceedings 1997 Workshop on Self-Organizing Maps, Helsinki University of Technology, Finland.