Vocabulary for Universal Approximation: A Linguistic Perspective of Mapping Compositions

TL;DR

This paper proves the existence of a finite set of mappings, or vocabulary, that can compose to approximate any continuous function on a compact domain, revealing new insights into the power of composition in neural networks.

Contribution

It constructs a finite vocabulary of mappings with size O(d^2) that can universally approximate any continuous function through composition, bridging deep learning and linguistic structures.

Findings

Finite vocabulary of size O(d^2) suffices for universal approximation.

Any continuous function can be approximated by compositions of vocabulary elements.

Results suggest a new perspective on the expressive power of compositional models.

Abstract

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.