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.

翻译：通用逼近定理表明，具有单隐藏层的神经网络能够在紧集上以任意期望精度逼近连续函数。该定理为神经网络在回归、分类等各类应用中的运用提供了理论支撑。此外，该定理适用于实数值神经网络及部分超复数值神经网络，如复值、四元数值、特瑟林值、克利福德值神经网络。然而，超复数值神经网络本质上是一种定义在具备额外代数或几何性质的代数结构上的向量值神经网络。本文扩展了通用逼近定理的适用范围，使其涵盖包括超复数值模型作为特例的广泛向量值神经网络。具体而言，我们引入非退化代数的概念，并阐述了定义在此类代数上的神经网络的通用逼近定理。