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Provable wavelet-based neural approximation

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  • Hur, Youngmi
  • Lim, Hyojae
  • Lim, Mikyoung

Abstract

In this paper, we develop a wavelet-based theoretical framework for analyzing the universal approximation capabilities of neural networks over a wide range of activation functions. Leveraging wavelet frame theory on the spaces of homogeneous type, we derive sufficient conditions on activation functions to ensure that the associated neural network approximates any functions in the function space induced by the activation function, along with an error estimate. These sufficient conditions accommodate a variety of smooth activation functions, including those that exhibit oscillatory behavior. Furthermore, by considering the L2-distance between smooth and non-smooth activation functions, we establish a generalized approximation result that is applicable to non-smooth activations, with the error explicitly controlled by this distance. This provides increased flexibility in the design of network architectures.

Suggested Citation

  • Hur, Youngmi & Lim, Hyojae & Lim, Mikyoung, 2026. "Provable wavelet-based neural approximation," Applied Mathematics and Computation, Elsevier, vol. 514(C).
    • Handle: RePEc:eee:apmaco:v:514:y:2026:i:c:s0096300325005466
    DOI: 10.1016/j.amc.2025.129821
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    References listed on IDEAS

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    1. Kadak, Ugur & Costarelli, Danilo & Coroianu, Lucian, 2023. "Neural network operators of generalized fractional integrals equipped with a vector-valued function," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
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