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

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Md. Haider Ali Biswas , Peter Massopust , R. Ramakrishnan ·

YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2408.06757

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 $\fl=\{f_1,\cdots, f_m\}\subset L^2(\R^n)$, we obtain $Φ=\{ϕ_1,\cdots,ϕ_\ell\}$ such that \[ V_θ(Φ)=\argmin\sum_{j=1}^m \|f_j-P_{V}f_j\|^2, \] where the minimum is taken over all $θ$-shift invariant spaces generated by at most $\ell$ 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.

Related Topics

Convolutional Neural Network

Fourier Transform

Computer Science

Artificial Intelligence

Algorithm

Mathematics

Fourier Analysis

Mathematical Analysis

Concepts

Convolutional neural network Fourier transform Fractional Fourier transform Computer science Discrete Fourier transform (general) Artificial intelligence Algorithm Pattern recognition (psychology) Mathematics Fourier analysis Mathematical analysis

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2408.06757
PDF: https://arxiv.org/pdf/2408.06757
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Related Works: 10
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DOI: https://doi.org/10.48550/arxiv.2408.06757

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

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Language: en

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Publication year: 2024

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Publication date: 2024-08-13

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Authors: Md. Haider Ali Biswas, Peter Massopust, R. Ramakrishnan

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Landing page: https://arxiv.org/abs/2408.06757

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Concepts: Convolutional neural network, Fourier transform, Fractional Fourier transform, Computer science, Discrete Fourier transform (general), Artificial intelligence, Algorithm, Pattern recognition (psychology), Mathematics, Fourier analysis, Mathematical analysis

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