Universal Approximation Theorem for Vector- and Hypercomplex-Valued
  Neural Networks
Marcos Eduardo Valle Wington L. Vital Guilherme Vieira
The universal approximation theorem states that a neural network with one
hidden layer can approximate continuous functions on compact sets with any
desired precision. This theorem supports using neural networks for various
applications, including regression and classification tasks. Furthermore, it is
valid for real-valued neural networks and some hypercomplex-valued neural
networks such as complex-, quaternion-, tessarine-, and Clifford-valued neural
networks. However, hypercomplex-valued neural networks are a type of
vector-valued neural network defined on an algebra with additional algebraic or
geometric properties. This paper extends the universal approximation theorem
for a wide range of vector-valued neural networks, including
hypercomplex-valued models as particular instances. Precisely, we introduce the
concept of non-degenerate algebra and state the universal approximation theorem
for neural networks defined on such algebras.