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

TL;DR

This paper introduces a deep convolutional neural network linked with the fractional Fourier transform and explores its properties, along with data approximation techniques in the FrFT domain, highlighting their theoretical foundations and potential applications.

Contribution

It defines a new neural network architecture based on the FrFT and studies its invariance properties, while also developing a data approximation framework in the FrFT domain with existence proofs.

Findings

The network is not translation invariant unlike classical networks.

Existence of a bandlimited function space closest to given data in the FrFT domain.

Optimal approximation spaces are characterized for data sets in L^2(R^n).

Abstract

In the first part of this paper, we define a deep convolutional neural network connected with the fractional Fourier transform (FrFT) using the $\theta$-translation operator, the translation operator associated with the FrFT. Subsequently, we study $\theta$-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 $\Phi=\{\phi_1,\cdots,\phi_\ell\}$ such that \[ V_\theta(\Phi)=\argmin\sum_{j=1}^m \|f_j-P_{V}f_j\|^2, \] where the minimum is taken over all $\theta$-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.