Computing With Contextual Numbers

Self Organizing Map has been applied into several classical modeling tasks such as clustering, classification, function approximation and visualization of high dimensional spaces. The final products of a trained SOM are a set of ordered (low dimensional) indices and their associated high dimensional weight vectors. While in all of the above-mentioned applications, the final high dimensional weight vectors play the main role in the computational steps, from a certain perspective, one can interpret SOM as a nonparametric encoder, in which the final low dimensional indices of the trained SOM are pointers to the high dimensional space. In this work, we showed, using a one-dimensional SOM, one can develop a nonparametric mapping from a high dimensional space to a continuous one dimensional numerical field, while the final values that we call contextual numbers are ordered in a way that in the given context, similar numbers refer to similar high dimensional states. As these numbers can be treated similarly to usual numbers, they can be embedded in many data driven modeling problems such as multimodal learning and data fusion. As a potential application, we showed how contextual numbers could be used for the problem of high dimensional spatiotemporal dynamics.