  We show that the core reasons that complex and hypercomplex valued neural
networks offer improvements over their real-valued counterparts is the weight
sharing mechanism and treating multidimensional data as a single entity. Their
algebra linearly combines the dimensions, making each dimension related to the
others. However, both are constrained to a set number of dimensions, two for
complex and four for quaternions. Here we introduce novel vector map
convolutions which capture both of these properties provided by
complex/hypercomplex convolutions, while dropping the unnatural dimensionality
constraints they impose. This is achieved by introducing a system that mimics
the unique linear combination of input dimensions, such as the Hamilton product
for quaternions. We perform three experiments to show that these novel vector
map convolutions seem to capture all the benefits of complex and hyper-complex
networks, such as their ability to capture internal latent relations, while
avoiding the dimensionality restriction.
