Uniqueness theorem for quaternionic neural networks

Quaternion algebra is a four-dimensional extension of the complex number field. The construction of artificial neural networks using complex numbers has recently been extended to that using quaternions. However, there have been few theoretical results for quaternionic neural networks. In the present work, we prove the applicability of the uniqueness theorem to quaternionic neural networks. Uniqueness theorems are important theories related to the singularities of neural networks. We provide the quaternionic versions of several important ideas, such as reducibility and equivalence, for proof of the uniqueness theorem. We can determine all the irreducible quaternionic neural networks that are I/O-equivalent to a given irreducible quaternionic neural network due to the uniqueness theorem.