Generalizing Complex/Hyper-complex Convolutions to Vector Map Convolutions

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.