Approximation of Time-Frequency Shift Equivariant Maps by Neural Networks

Dae Gwan Lee ()
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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

Mathematics, 2024, vol. 12, issue 23, 1-18

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.

Keywords: neural networks; equivariance; time-frequency shifts; time-frequency analysis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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