Vocabulary for Universal Approximation: A Linguistic Perspective of Mapping Compositions
Primary Area: general machine learning (i.e., none of the above)
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Keywords: Neural Networks, Approximation Theory, Composite Mappings, Sequence Modeling
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TL;DR: Achieve universal approximation by compositing a sequence of mappings in a finite set, similar to assembling words into sentences in linguistics.
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 $\phi_{i_1}, ..., \phi_{i_m} \in V, m \in Z_+$, 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.
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Submission Number: 4427
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