The field of deep learning has seen significant advancement in recent years. However, much of the existing work has been focused on real-valued numbers. Recent work has shown that a deep learning system using the complex numbers can be deeper for a fixed parameter budget compared to its real-valued counterpart. In this work, we explore the benefits of generalizing one step further into the hyper-complex numbers, quaternions specifically, and provide the architecture components needed to build deep quaternion networks. We develop the theoretical basis by reviewing quaternion convolutions, developing a novel quaternion weight initialization scheme, and developing novel algorithms for quaternion batch-normalization. These pieces are tested in a classification model by end-to-end training on the CIFAR-10 and CIFAR-100 data sets and a segmentation model by end-to-end training on the KITTI Road Segmentation data set. These quaternion networks show improved convergence compared to real-valued and complex-valued networks, especially on the segmentation task, while having fewer parameters.

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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. 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 this property, 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 two experiments to show that these novel vector map convolutions seem to capture all the benefits of complex and hyper-complex networks, while avoiding the dimensionality restriction.