Radosław Antoni Kycia

Agnieszka Niemczynowicz, Radosław Antoni Kycia

Neural networks used in computations with more advanced algebras than real numbers perform better in some applications. However, there is no general framework for constructing hypercomplex neural networks. We propose a library integrated with Keras that can do computations within TensorFlow and PyTorch. It provides Dense and Convolutional 1D, 2D, and 3D layers architectures.

Agnieszka Niemczynowicz, Radosław Antoni Kycia

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.