Removing Dimensional Restrictions on Complex/Hyper-complex Convolutions

It has been shown that the core reasons that complex and hypercomplex valued neural networks offer improvements over their real-valued counterparts is the fact that aspects of their algebra forces treating multi-dimensional data as a single entity (forced local relationship encoding) with an added benefit of reducing parameter count via weight sharing. However, both are constrained to a set number of dimensions, two for complex and four for quaternions. These observations motivate us to introduce novel vector map convolutions which capture both of these properties provided by complex/hypercomplex convolutions, while dropping the unnatural dimensionality constraints their algebra imposes. This is achieved by introducing a system that mimics the unique linear combination of input dimensions via the Hamilton product using a permutation function, as well as batch normalization and weight initialization for the system. We perform three experiments using three different network architectures 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.