Deep convolutional neural networks and data approximation using the fractional Fourier transform

2024/08/13 by Md. Haider Ali Biswas, Biswas, M. H. A., Peter Massopust +3 · 1 citation
Computer Science · Engineering · #Image and Signal Denoising Methods #Ultrasonics and Acoustic Wave Propagation

paper · pdf · doi:10.48550/arxiv.2408.06757

Abstract

In the first part of this paper, we define a deep convolutional neural network connected with the fractional Fourier transform (FrFT) using the θ-translation operator, the translation operator associated with the FrFT. Subsequently, we study θ-translation invariance properties of this network. Unlike the classical case, these networks are not translation invariant. \par In the second part, we study data approximation problems using the FrFT. More precisely, given a data set \fl=\f₁,⋯, f_m\⊂ L²(\Rⁿ), we obtain Φ=\ϕ₁,⋯,ϕ_ℓ\ such that V_θ(Φ)=\argmin∑_j=1^m ‖f_j-P_Vf_j‖², where the minimum is taken over all θ-shift invariant spaces generated by at most ℓ elements. Moreover, we prove the existence of a space of bandlimited functions in the FrFT domain which is ``closest" to \fl in the above sense.

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