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Approximation of Time-Frequency Shift Equivariant Maps by Neural Networks

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  • Dae Gwan Lee

    (Department of Mathematics and Big Data Science, Kumoh National Institute of Technology, Gumi 39177, Gyeongsangbuk-do, Republic of Korea)

Abstract

Based on finite-dimensional time-frequency analysis, we study the properties of time-frequency shift equivariant maps that are generally nonlinear. We first establish a one-to-one correspondence between Λ -equivariant maps and certain phase-homogeneous functions and also provide a reconstruction formula that expresses Λ -equivariant maps in terms of these phase-homogeneous functions, leading to a deeper understanding of the class of Λ -equivariant maps. Next, we consider the approximation of Λ -equivariant maps by neural networks. In the case where Λ is a cyclic subgroup of order N in Z N × Z N , we prove that every Λ -equivariant map can be approximated by a shallow neural network whose affine linear maps are simply linear combinations of time-frequency shifts by Λ . This aligns well with the proven suitability of convolutional neural networks (CNNs) in tasks requiring translation equivariance, particularly in image and signal processing applications.

Suggested Citation

  • Dae Gwan Lee, 2024. "Approximation of Time-Frequency Shift Equivariant Maps by Neural Networks," Mathematics, MDPI, vol. 12(23), pages 1-18, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3704-:d:1530019
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    References listed on IDEAS

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    1. David Silver & Aja Huang & Chris J. Maddison & Arthur Guez & Laurent Sifre & George van den Driessche & Julian Schrittwieser & Ioannis Antonoglou & Veda Panneershelvam & Marc Lanctot & Sander Dieleman, 2016. "Mastering the game of Go with deep neural networks and tree search," Nature, Nature, vol. 529(7587), pages 484-489, January.
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