Geometric Feedforward Neural Networks and Support Multivector Machines

The representation of the external world in biological creatures appears to be definable in terms of geometry. We can formalize the relationships between the physical signals of external objects and the internal signals of a biological creature by using extrinsic vectors coming from the world and intrinsic vectors representing the world internaly. We can also assume that the external world and the internal world have different reference coordinate systems. If we consider the acquisition and coding of knowledge as a distributed and differentiated process, it is imaginable that there should e-xist various domains of knowledge representation obeying different metrics which can be modelled using different vectorial basis. How it is possible that nature could have acquired through evolution such tremendous representation power for dealing with complicated geometric signal processing [ 13 ]? Pellionisz and Llinàs [ 15 , 16 ] claim in a stimulating series of articles that the formalization of the geometrical representation seems to be the dual expression of extrinsic physical cues performed by the intrinsic central nervous system vectors. These vectorial representations, related to reference frames intrinsic to the creature, are covariant for perception analysis and contravariant for action synthesis. The authors explain that the geometric mapping between these two vectorial spaces can be implemented by a neural network which performs as a metric tensor [ 16 ].