扩展一大类超复数值神经网络的万能逼近定理
Extending the Universal Approximation Theorem for a Broad Class of Hypercomplex-Valued Neural Networks
AI总结：
本文通过引入非退化超复代数概念，将万能逼近定理扩展至一大类超复数值神经网络，涵盖复数、四元数和tessarines等代数，为该类网络的存在性提供了理论支撑。
摘要：
万能逼近定理断言，单隐藏层神经网络能在紧致集上以任意所需精度逼近连续函数。作为一种存在性结果，万能逼近定理支持了神经网络在包括回归和分类任务在内的各种应用中的使用。万能逼近定理不仅限于实值神经网络，也适用于复数、四元数、tessarines 和 Clifford 值神经网络。本文将万能逼近定理扩展到了一大类超复数值神经网络。具体而言，我们首先引入了非退化超复代数的概念。复数、四元数和 tessarines 都是非退化超复代数的例子。然后，我们陈述了定义在非退化代数上的超复数值神经网络的万能逼近定理。
英文摘要：
The universal approximation theorem asserts that a single hidden layer neural network approximates continuous functions with any desired precision on compact sets. As an existential result, the universal approximation theorem supports the use of neural networks for various applications, including regression and classification tasks. The universal approximation theorem is not limited to real-valued neural networks but also holds for complex, quaternion, tessarines, and Clifford-valued neural networks. This paper extends the universal approximation theorem for a broad class of hypercomplex-valued neural networks. Precisely, we first introduce the concept of non-degenerate hypercomplex algebra. Complex numbers, quaternions, and tessarines are examples of non-degenerate hypercomplex algebras. Then, we state the universal approximation theorem for hypercomplex-valued neural networks defined on a non-degenerate algebra.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
