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Spotlight Poster

Vocabulary for Universal Approximation: A Linguistic Perspective of Mapping Compositions

Yongqiang Cai

Hall C 4-9 #2000

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 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=ϕi:RdRd|i=1,...,n with n=O(d2) for the universal approximation. That is, for any continuous mapping f:RdRd, compact domain Ω and ε>0, there is a sequence of mappings ϕi1,...,ϕimV,mZ+, such that the composition ϕim...ϕi1 approximates f on Ω with an error less than ε. Our results demonstrate an unusual approximation power of mapping compositions and motivate a novel compositional model for regular languages.

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