In recent years, deep learning-based sequence modelings, such as language models, have received much attention and success, which pushes researchers to explore the possibility of transforming non-sequential problems into a sequential form. Following this thought, deep neural networks can be represented as composite functions of a sequence of mappings, linear or nonlinear, where each composition can be viewed as a \emph{word}. However, the weights of linear mappings are undetermined and hence require an infinite number of words. In this article, we investigate the finite case and constructively prove the existence of a finite \emph{vocabulary} $V=\{\phi_i: \mathbb{R}^d \to \mathbb{R}^d | i=1,...,n\}$ with $n=O(d^2)$ for the universal approximation. That is, for any continuous mapping $f: \mathbb{R}^d \to \mathbb{R}^d$, compact domain $\Omega$ and $\varepsilon>0$, there is a sequence of mappings $\phi_{i_1}, ..., \phi_{i_m} \in V, m \in \mathbb{Z}_+$, such that the composition $\phi_{i_m} \circ ... \circ \phi_{i_1} $ approximates $f$ on $\Omega$ with an error less than $\varepsilon$. Our results demonstrate an unusual approximation power of mapping compositions and motivate a novel compositional model for regular languages.

翻译：近年来，基于深度学习的序列建模（如语言模型）获得了广泛关注与成功，这促使研究者探索将非序列问题转化为序列形式的可能性。沿此思路，深度神经网络可表示为一系列线性或非线性映射的复合函数，其中每个组合可视为一个\emph{词}。然而线性映射的权重未确定，因此需要无限数量的词汇。本文研究有限情形，通过构造性方法证明存在一个有限\emph{词汇表}$V=\{\phi_i: \mathbb{R}^d \to \mathbb{R}^d | i=1,...,n\}$（其中$n=O(d^2)$）可实现通用逼近。即对于任意连续映射$f: \mathbb{R}^d \to \mathbb{R}^d$、紧致域$\Omega$及$\varepsilon>0$，总存在映射序列$\phi_{i_1}, ..., \phi_{i_m} \in V, m \in \mathbb{Z}_+$，使得复合映射$\phi_{i_m} \circ ... \circ \phi_{i_1}$在$\Omega$上以小于$\varepsilon$的误差逼近$f$。我们的结果揭示了映射组合非凡的逼近能力，并为正则语言的新型组合模型提供了理论依据。