Chase J. Gaudet

Chase J Gaudet, Anthony S Maida

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.

Chase Gaudet, Anthony Maida

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