A fully tensorial theoretical framework for hypercomplex-valued neural networks is presented. The proposed approach enables neural network architectures to operate on data defined over arbitrary finite-dimensional algebras. The central observation is that algebra multiplication can be represented by a rank-three tensor, which allows all algebraic operations in neural network layers to be formulated in terms of standard tensor contractions, permutations, and reshaping operations. This tensor-based formulation provides a unified and dimension-independent description of hypercomplex-valued dense and convolutional layers and is directly compatible with modern deep learning libraries supporting optimized tensor operations. The proposed framework recovers existing constructions for four-dimensional algebras as a special case. Within this setting, a tensor-based version of the universal approximation theorem for single-layer hypercomplex-valued perceptrons is established under mild non-degeneracy assumptions on the underlying algebra, thereby providing a rigorous theoretical foundation for the considered class of neural networks.

翻译：本文提出了一种用于超复数神经网络的完全张量理论框架。所提出的方法使神经网络架构能够在任意有限维代数定义的数据上运行。核心观察在于代数乘法可由一个三阶张量表示，这使得神经网络层中的所有代数运算都能通过标准张量缩并、置换和重塑操作来表述。这种基于张量的表述为超复数全连接层和卷积层提供了统一且维度无关的描述，并可直接兼容支持优化张量运算的现代深度学习库。该框架将四维代数的现有构造作为特例进行恢复。在此设定下，基于底层代数的温和非退化假设，建立了单层超复数感知机的通用逼近定理的张量形式，从而为所考虑的神经网络类别提供了严格的理论基础。