Deep fractional Fourier networks

Abstract

This thesis introduces the integration of the fractional Fourier Transform (FrFT) into the deep learning domain, with the aim of opening new avenues for incorporating signal processing into deep neural networks (DNNs). This work starts by introducing FrFT into recurrent neural networks (RNNs) for time series prediction, leveraging its ability and flexibility to perform infinitely many continuous transformations and offering an alternative to the traditional Fourier Transform (FT). Despite the initial success, a significant challenge identified is the manual tuning of the fraction order parameter a, which can be cumbersome and limits broader applicability. To overcome this limitation, we introduce a novel approach where the fraction order a is treated as a learnable parameter within deep learning models. First, a theoretical foundation is established to support the learnability of this parameter, followed by extensive experimentation in image classification and time series prediction tasks. The results demonstrate that incorporating a learnable fraction order significantly improves model performance, particularly when integrated with well-known architectures such as ResNet and VGG models. Furthermore, the thesis proposes fractional Fourier Pooling (FrFP), a pooling technique that replaces traditional Global Average Pooling (GAP) layers in Convolutional Neural Networks (CNNs). FrFP enhances feature representation by processing intermediate signal regions, leading to better model performance and offering a new perspective on integrating signal transformations within deep learning frameworks. Overall, this thesis contributes to the growing body of research exploring advanced signal processing techniques in deep learning, highlighting the potential of FrFT as a powerful tool for improving model accuracy and efficiency across various applications.