Vocabulary for Universal Approximation: A Linguistic Perspective of Mapping Compositions
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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 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 vocabulary V={ϕ_i: ℝ^d →ℝ^d | i=1,...,n} with n=O(d^2) for the universal approximation. That is, for any continuous mapping f: ℝ^d →ℝ^d, compact domain Ω and ε>0, there is a sequence of mappings ϕ_i_1, ..., ϕ_i_m∈ V, m ∈ℤ_+, such that the composition ϕ_i_m∘ ... ∘ϕ_i_1 approximates f on Ω with an error less than ε. Our results provide a linguistic perspective of composite mappings and suggest a cross-disciplinary study between linguistics and approximation theory.
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