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In this article, we develop a new method to approximate numerically the fractional Laplacian of functions defined on $\mathbb R$, as well as some more general singular integrals. After mapping $\mathbb R$ into a finite interval, we…
We present a Fourier neural network (FNN) that can be mapped directly to the Fourier decomposition. The choice of activation and loss function yields results that replicate a Fourier series expansion closely while preserving a…
Fourier transform (FT) plays a crucial role in a broad range of applications, from enhancement, restoration and analysis through to security, compression and manipulation. The Fourier transform (FT) is a process that converts a function…
We address the problem of decomposing an image into albedo and shading. We propose the Fast Fourier Intrinsic Network, FFI-Net in short, that operates in the spectral domain, splitting the input into several spectral bands. Weights in…
This paper explores the innovative application of the Fractional Fourier Transform (FrFT) in sound synthesis, highlighting its potential to redefine time-frequency analysis in audio processing. As an extension of the classical Fourier…
Deep learning based on deep neural networks of various structures and architectures has been powerful in many practical applications, but it lacks enough theoretical verifications. In this paper, we consider a family of deep convolutional…
In recent years there has been a growing interest in the fractional Fourier transform driven by its large number of applications. The literature in this field follows two main routes. On the one hand, the areas where the ordinary Fourier…
Deep convolutional neural networks have led to breakthrough results in numerous practical machine learning tasks such as classification of images in the ImageNet data set, control-policy-learning to play Atari games or the board game Go,…
Deep-learning density functional theory (DFT) shows great promise to significantly accelerate material discovery and potentially revolutionize materials research. However, current research in this field primarily relies on data-driven…
Currently, it is hard to reap the benefits of deep learning for Bayesian methods, which allow the explicit specification of prior knowledge and accurately capture model uncertainty. We present Prior-Data Fitted Networks (PFNs). PFNs…
This paper introduces a hybrid computational framework for the multi-frequency inverse source problem governed by the Helmholtz equation. By integrating a classical Fourier method with a deep convolutional neural network, we address the…
Invariance to spatial transformations such as translations and rotations is a desirable property and a basic design principle for classification neural networks. However, the commonly used convolutional neural networks (CNNs) are actually…
This brief note aims at condensing some results on the 32-point approximate DFT and discussing its arithmetic complexity.
Transformers have captured growing attention in computer vision, thanks to its large capacity and global processing capabilities. However, transformers are data hungry, and their ability to generalize is constrained compared to…
Large language models have transformed natural language processing, yet supervised fine-tuning (SFT) remains computationally intensive. This paper formally proves that capabilities acquired through SFT can be approximated by a base…
In this paper, we aim at establishing an approximation theory and a learning theory of distribution regression via a fully connected neural network (FNN). In contrast to the classical regression methods, the input variables of distribution…
In this letter, a fast Fourier transform (FFT)-enhanced low-complexity super-resolution sensing algorithm for near-field source localization with both angle and range estimation is proposed. Most traditional near-field source localization…
Fast Fourier convolution (FFC) is the recently proposed neural operator showing promising performance in several computer vision problems. The FFC operator allows employing large receptive field operations within early layers of the neural…
We discuss the advantages of using the approximate quantum Fourier transform (AQFT) in algorithms which involve periodicity estimations. We analyse quantum networks performing AQFT in the presence of decoherence and show that extensive…
The graph Hilbert transform (GHT) is a key tool in constructing analytic signals and extracting envelope and phase information in graph signal processing. However, its utility is limited by confinement to the graph Fourier domain, a fixed…