[joe_the_user]

A nice, cogent explanation.

It's good to remember the ANN's input offset comes as vector data. The ANN isn't directly transforming those vectors directly, rather it is transforms these input to a higher dimensional "feature" space and performs the linear transform. If you take the separating plane that's drawn in the feature space and reverse the map, you'll the ANN has drawn complex surface between the points it want to recognize and those it rejects.

So it's basically a heuristic and no more intelligent than Taylor's series.