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 provide a linguistic perspective of composite mappings and suggest a cross-disciplinary study between linguistics and approximation theory.

翻译：近期，基于深度学习的序列建模（如语言模型）备受关注并取得了显著成功，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。遵循这一思路，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，其中每个复合可视为一个“词”。然而，线性映射的权重尚未确定，因此需要无限数量的词。本文研究有限情形，并构造性地证明存在一个有限“词汇表”$V=\{\phi_i: \mathbb{R}^d \to \mathbb{R}^d | i=1,...,n\}$（其中$n=O(d^2)$）可用于通用逼近。即，对于任意连续映射$f: \mathbb{R}^d \to \mathbb{R}^d$、紧致域$\Omega$和$\varepsilon>0$，存在一个映射序列$\phi_{i_1}, ..., \phi_{i_m} \in V, m \in \mathbb{Z}_+$，使得复合映射$\phi_{i_m} \circ ... \circ \phi_{i_1}$在$\Omega$上逼近$f$且误差小于$\varepsilon$。我们的结果为复合映射提供了语言学视角，并提出了语言学与逼近理论之间的跨学科研究。