Extreme Learning Machines on Cayley-Dickson Algebra Applied for Color Image Auto-Encoding

This paper aims to provide a useful framework for extreme learning machines (ELMs) on Cayley-Dickson algebras. Cayley-Dickson algebras, which include complex numbers, quaternions, and octonions as particular instances, are hyper-complex algebras defined using a recursive procedure. Firstly, we review some basic concepts on Cayley-Dickson algebras and formulate Cayley-Dickson matrix product using real-valued linear algebra. Then, we propose the Cayley-Dickson ELMs and derive their learning using Cayley-Dickson least squares problem. Lastly, we compare the performance of real-valued and four-dimensional Cayley-Dickson ELM models, including quaternion-valued ELM, in an experiment on color image auto-encoding using the well-known CIFAR dataset.