Fourier Basis Research Articles

On an Exact Convergence of Quasi-Periodic Interpolations for the Polyharmonic–Neumann Eigenfunctions

Fourier expansions employing polyharmonic–Neumann eigenfunctions have demonstrated improved convergence over those using the classical trigonometric system, due to the rapid decay of their Fourier coefficients. Building on this insight, we investigate interpolations on a finite interval that are exact for polyharmonic–Neumann eigenfunctions and exhibit similar benefits. Furthermore, we enhance the convergence of these interpolations by incorporating the concept of quasi-periodicity, wherein the basis functions are periodic over a slightly extended interval. We demonstrate that those interpolations achieve significantly better convergence rates away from the endpoints of the approximation interval and offer increased accuracy over the entire interval. We establish these properties for a specific case of polyharmonic–Neumann eigenfunctions known as the modified Fourier system. For other basis functions, we provide supporting evidence through numerical experiments. While the latter methods display superior convergence rates, we demonstrate that interpolations using the modified Fourier basis offer distinct advantages. Firstly, they permit explicit representations via the inverses of certain Vandermonde matrices, whereas other interpolation methods require approximate computations of the eigenvalues and eigenfunctions involved. Secondly, these matrix inverses can be efficiently computed for numerical applications. Thirdly, the introduction of quasi-periodicity improves the convergence rates, making them comparable to those of other interpolation techniques.

Extraordinarily Time- and Memory-Efficient Large-Scale Canonical Correlation Analysis in Fourier Domain: From Shallow to Deep.

Canonical correlation analysis (CCA) is a correlation analysis technique that is widely used in statistics and the machine-learning community. However, the high complexity involved in the training process lays a heavy burden on the processing units and memory system, making CCA nearly impractical in large-scale data. To overcome this issue, a novel CCA method that tries to carry out analysis on the dataset in the Fourier domain is developed in this article. Appling Fourier transform on the data, we can convert the traditional eigenvector computation of CCA into finding some predefined discriminative Fourier bases that can be learned with only element-wise dot product and sum operations, without complex time-consuming calculations. As the eigenvalues come from the sum of individual sample products, they can be estimated in parallel. Besides, thanks to the data characteristic of pattern repeatability, the eigenvalues can be well estimated with partial samples. Accordingly, a progressive estimate scheme is proposed, in which the eigenvalues are estimated through feeding data batch by batch until the eigenvalues sequence is stable in order. As a result, the proposed method shows its characteristics of extraordinarily fast and memory efficiencies. Furthermore, we extend this idea to the nonlinear kernel and deep models and obtained satisfactory accuracy and extremely fast training time consumption as expected. An extensive discussion on the fast Fourier transform (FFT)-CCA is made in terms of time and memory efficiencies. Experimental results on several large-scale correlation datasets, such as MNIST8M, X-RAY MICROBEAM SPEECH, and Twitter Users Data, demonstrate the superiority of the proposed algorithm over state-of-the-art (SOTA) large-scale CCA methods, as our proposed method achieves almost same accuracy with the training time of our proposed method being 1000 times faster. This makes our proposed models best practice models for dealing with large-scale correlation datasets. The source code is available at https://github.com/Mrxuzhao/FFTCCA.

Graph signal processing and graph learning approaches to Schizophrenia pattern identification in brain Electroencephalogram

The detection of Schizophrenia (SZ) directly from brain Electroencephalogram (EEG) signals has recently gained importance. Traditionally, EEG-based SZ detection is done by either manually extracting features from each EEG channel and using Machine Learning classifiers or by Deep Learning-based automated detection. These methods neglect the functional interactions between spatially distributed brain regions. This study proposes a graph signal processing and graph learning framework to detect SZ. Here, the individual channels’ features and the global interactions are combined into a unified graph signal (GS) representation. The proposed representation consists of the GS values and the underlying connectivity network. An SZ-specific feature, namely, the permutation entropies of each electrode’s signal, is considered as the GS values. The graph learning method learns the underlying network from a collection of GS observations. The learned network serves as the Fourier basis for the graph Fourier and wavelet transform. Utilizing these transforms, the graph spectral features are extracted from GSs. Lastly, the extracted features are classified using a set of Machine Learning models. The proposed graph signal processing and graph learning method is an effective approach to decode Schizophrenia patterns in brain EEG signals and outperform the existing approaches.

Shannon’s Sampling Theorem for Set-Valued Functions with an Application

In this study, we defined a kind of Fourier expansion of set-valued square-integrable functions. In fact, we have seen that the classical Fourier basis also constitutes a basis for the Hilbert quasilinear space L2(−π,π,Ω(C)) of Ω(C)-valued square-integrable functions, where Ω(C) is the class of all compact subsets of complex numbers. Furthermore, we defined the quasi-Paley–Wiener space, QPW, using the Fourier transform defined for set-valued functions and thus we showed that the sequence sinc.−kk∈Z form also a basis for QPW. We call this result Shannon’s sampling theorem for set-valued functions. Finally, we gave an application based on this theorem.

[formula omitted]-space physics-informed neural network (k-PINN) for compressed spectral mapping and efficient inversion of vibrations in thin composite laminates

The vibrational response of structural components carries valuable information about their underlying mechanical properties, health status and operational conditions. This underscores the need for the development of efficient physics-based inversion algorithms which, given a limited set of sensing data points and in the presence of measurement noise, can reconstruct the response at locations where measurement data is not available and/or identify the unknown mechanical properties. Addressing this challenge, Physics-Informed Neural Networks (PINNs) have emerged as a promising approach. PINNs seamlessly integrate governing equations into their architecture and have gained significant interest in solving inversion problems. In the context of learning and inversion of multimodal, multiscale vibrational responses, this paper introduces a novel spectral extension of PINNs, utilizing Fourier basis functions in the wavenumber domain, commonly known as k-space. The proposed method, referred to as k-space PINN (k-PINN), offers a robust framework for adjusting complexity and wavenumber composition of the response. Notably, the spectral formulation of k-PINN, coupled with the generally sparse representation of vibrations in k-space, facilitate efficient reconstruction and learning of broadband vibrations and alleviate the spectral bias associated with standard PINN. Additionally, the spectral solution space introduced by k-PINN substantially reduces the computational cost associated with computing physics-informed loss terms. We evaluate the effectiveness of the proposed methodology on reconstructing the bending vibrational mode shapes of a thin composite laminate and identifying its effective bending stiffness coefficients. Mode shapes are initially obtained from finite element simulation, and virtual test data with added noise are generated for evaluation purposes. It is shown that the proposed k-PINN methodology outperforms the standard PINN in terms of both learning and computational efficiency. The performance of k-PINN is further demonstrated by show casing its capability in learning different selections of symmetric, anti-symmetric and asymmetric mode shapes.

Hermitian random walk graph Fourier transform for directed graphs and its applications

Signal processing on directed graphs present additional challenges since a complete set of eigenvectors is unavailable generally. To solve this problem, in this paper, a novel graph Fourier transform is constructed for representing and processing signals on directed graphs. Firstly, we introduce a Hermitian random walk Laplacian operator and derive that it is Hermitian positive semi-definite. Hence, the obtained Laplacian operator is diagonalizable and yields orthogonal eigenvectors as graph Fourier basis. Secondly, we propose the Hermitian random walk graph Fourier transform (HRWGFT) with good properties including unitary and preserving inner products. Furthermore, HRWGFT records the directionality of edges without sacrificing the information about the graph signal. Then, using these favorable properties, we derive spectral convolution to define the graph filter which is the core tool for processing graph signals. Finally, based on the proposed Laplacian matrix and HRWGFT, we present several applications on synthetic and real-world networks, including signal denoising, data classification. The rationality and validity of our work are verified by simulations.

Adaptive sampling strategy for Fourier single-pixel imaging

Fourier single-pixel imaging is a technique that utilizes Fourier bases as illumination patterns to capture images. While the use of orthogonal bases enhances the imaging quality compared with other methods of single pixel imaging, the slow imaging speed has hindered its broader application. Acquiring only low-frequency Fourier coefficients enables capturing a significant portion of image information at low sampling ratio, thereby enhancing imaging speed. Nevertheless, truncating frequencies results in a loss of image detail and a decrease in picture quality. To address this challenge, this paper introduces an adaptive sampling strategy for Fourier single-pixel imaging, allowing for customized sampling distributions for each specific object. The sampling distribution is dynamically generated based on the acquired Fourier coefficients during the acquisition process, eliminating the need for prior information. Through extensive simulations and experimental analysis, this approach has demonstrated its ability to capture more complex image details at equivalent sampling ratios.

Research on Fourier Coefficient-Based Energy Capture for Direct-Drive Wave Energy Generation System Based on Position Sensorless Disturbance Suppression

In order to improve the energy capture efficiency of direct-drive wave power generation (DDWEG) systems and enhance the robustness of the reference power tracking control, a Fourier coefficient-based energy capture (FCBEC) and a position sensorless disturbance suppression (PSDS) control strategy are proposed. For energy capture, FCBEC is proposed to construct the objective function by maximizing the average power over a period of time and expanding the variables in the Fourier basis when the maximum power is captured, which is used as the basis for obtaining the reference trajectory. To address the limitations of the mechanical encoder, the position sensorless technique, based on a sliding mode observer (SMO), is used in the power tracking control, and the position information is obtained through an inverse tangent function. The perturbation caused by the inverse electromotive force error in the system is theoretically analyzed. A full-order terminal sliding mode approach is employed to design a current controller that suppresses the perturbation and ensures accurate tracking of the reference current. Simulation results show that the ocean-wave energy capture strategy proposed in this paper can make the energy captured by the PTO reach the optimal value under the impedance matching condition, and that the response speed and robustness of the full-order terminal sliding mode are better than the traditional PI control.

Low-sampling high-quality Hadamard and Fourier single-pixel imaging through automated optimization neural network

In this manuscript, an automated optimization neural network is applied in Hadamard single-pixel imaging (H-SPI) and Fourier single-pixel imaging (F-SPI) to improve the imaging quality at low sampling ratios which is called AO-Net. By projecting Hadamard or Fourier basis illumination light fields onto the object, a single-pixel detector is used to collect the reflected light intensities from object. The one-dimensional detection values are fed into the designed AO-Net, and the network can automatically optimize. Finally, high-quality images are output through multiple iterations without pre-training and datasets. Numerical simulations and experiments demonstrate that AO-Net outperforms other existing widespread methods for both binary and grayscale images at low sampling ratios. Specially, the Structure Similarity Index Measure value of the binary reconstructed image can reach more than 0.95 when the sampling ratio is less than 3%. Therefore, AO-Net holds great potential for applications in the fields of complex environment imaging and moving object imaging.

Spectral Galerkin mode-matching method for applications in photonics.

Many engineered photonic devices can be decomposed into parts where the material properties are independent of one or more spatial variables. Numerical mode-matching methods are widely used to simulate such photonic devices due to the efficiency gained by treating the separated variables analytically. Existing mode-matching methods based on piecewise polynomials are more accurate than those based on the global Fourier basis or low-order finite difference, finite-element schemes, but they may exhibit numerical instability when a large number of eigenmodes are used. To overcome this difficulty, we introduce the spectral Galerkin mode matching method (SGMM) based on a global piecewise-polynomial basis and a Galerkin method to solve the eigenmodes. It is shown that the numerical eigenmodes of SGMM preserve the pseudo-orthogonality of the analytical eigenmodes. This property leads to linear systems that are typically well-conditioned. Numerical examples indicate that SGMM is more stable than other mode matching methods, and gives reliable results even when a large number of eigenmodes are used.

Differentially private federated learning with Laplacian smoothing

Federated learning aims to protect data privacy by collaboratively learning a model without sharing private data among users. However, an adversary may still be able to infer the private training data by attacking the released model. Differential privacy provides a statistical protection against such attacks at the price of significantly degrading the accuracy or utility of the trained models. In this paper, we investigate a utility enhancement scheme based on Laplacian smoothing for differentially private federated learning (DP-Fed-LS), to improve the statistical precision of parameter aggregation with injected Gaussian noise without losing privacy budget. Our key observation is that the aggregated gradients in federated learning often enjoy a type of smoothness, i.e. sparsity in a graph Fourier basis with polynomial decays of Fourier coefficients as frequency grows, which can be exploited by the Laplacian smoothing efficiently. Under a prescribed differential privacy budget, convergence error bounds with tight rates are provided for DP-Fed-LS with uniform subsampling of heterogeneous non-iid data, revealing possible utility improvement of Laplacian smoothing in effective dimensionality and variance reduction, among others. Experiments over MNIST, SVHN, and Shakespeare datasets show that the proposed method can improve model accuracy with DP-guarantee and membership privacy under both uniform and Poisson subsampling mechanisms.

ANOVA approximation with mixed tensor product basis on scattered points

In this paper we consider an orthonormal basis, generated by a tensor product of Fourier basis functions, half period cosine basis functions, and the Chebyshev basis functions. We deal with the approximation problem in high dimensions related to this basis and design a fast algorithm to multiply with the underlying matrix, consisting of rows of the non-equidistant Fourier matrix, the non-equidistant cosine matrix and the non-equidistant Chebyshev matrix, and its transposed. Using this, we derive the ANOVA (analysis of variance) decomposition for functions with partially periodic boundary conditions through using the Fourier basis in some dimensions and the half period cosine basis or the Chebyshev basis in others. We consider sensitivity analysis in this setting, in order to find an adapted basis for the underlying approximation problem. More precisely, we find the underlying index set of the multidimensional series expansion. Additionally, we test this ANOVA approximation with mixed basis at numerical experiments, and refer to the advantage of interpretable results.

Aliasing from Galactic Plane Setting in Widefield Radio Interferometry

Measurements with widefield radio interferometers often include the near-infinite gradient between the sky and the horizon. This causes aliasing inherent to the measurement itself and is purely a consequence of the Fourier basis. For this reason, the horizon is often attenuated by the instrumental beam down to levels deemed inconsequential. However, this effect is enhanced via our own Galactic plane as it sets over the course of a night. We show all-sky simulations of the Galactic plane setting in a low-frequency radio interferometer in detail for the first time. We then apply these simulations to the Murchison Widefield Array to show that a beam attenuation of 0.1% is not sufficient in some precision science cases. We determine that the noise statistics of the residual data image are drastically more Gaussian with aliasing removal, and explore consequences in simulation for cataloging of extragalactic sources and 21 cm Epoch of Reionization detection via the power spectrum.

Nonadiabatic Conical Intersection Dynamics in the Local Diabatic Representation with Strang Splitting and Fourier Basis.

We develop and implement an exact conical intersection nonadiabatic wave packet dynamics method that combines the local diabatic representation, Strang splitting for the total molecular propagator, and discrete variable representation with uniform grids. By employing the local diabatic representation, this method captures all nonadiabatic effects, including nonadiabatic transitions, electronic coherences, and geometric phase. Moreover, it is free of singularities in the first and second derivative couplings and does not require the electronic wave function to be continuous with respect to the nuclear coordinates. We further show that in contrast to the adiabatic representation, the split-operator method can be directly applied to the full molecular propagator with the locally diabatic ansatz. The Fourier series, employed as the primitive nuclear basis functions, is universal and can be applied to all types of reactive coordinates. The combination of local diabatic representation, Strang splitting, and Fourier basis allows numerically exact modeling of conical intersection quantum dynamics directly with adiabatic electronic states that can be obtained from standard electronic structure computations.

Weighted Compressive Sensing Applied to Seismic Interferometry: Wavefield Reconstruction Using Prior Information

Abstract Seismic interferometry is widely used for passive subsurface investigation using seismic noise. The technique requires much storage for long noise records to suppress interferometric noise, which consists of spurious arrivals that do not correspond to the inter-receiver surface waves. Such long recordings may not be available in practice. Compressive sensing (CS), which is a wavefield reconstruction technique operating on incomplete data, may increase the availability, and reduce storage limitations of long noise time series. Using a numerical example of a linear array surrounded by sources and the Fourier basis for a sparse transform, we show that inter-receiver wavefields can be recovered at the locations where seismometers are unavailable, reducing the storage required for interferometry. We propose and develop a weighted CS algorithm that helps suppress the spurious arrivals by incorporating a priori information about the arrivals of surface waves that can be expected.

Learning Spatially Collaged Fourier Bases for Implicit Neural Representation

Existing approaches to Implicit Neural Representation (INR) can be interpreted as a global scene representation via a linear combination of Fourier bases of different frequencies. However, such universal basis functions can limit the representation capability in local regions where a specific component is unnecessary, resulting in unpleasant artifacts. To this end, we introduce a learnable spatial mask that effectively dispatches distinct Fourier bases into respective regions. This translates into collaging Fourier patches, thus enabling an accurate representation of complex signals. Comprehensive experiments demonstrate the superior reconstruction quality of the proposed approach over existing baselines across various INR tasks, including image fitting, video representation, and 3D shape representation. Our method outperforms all other baselines, improving the image fitting PSNR by over 3dB and 3D reconstruction to 98.81 IoU and 0.0011 Chamfer Distance.

Unveiling dynamics changes: Singular spectrum analysis-based method for detecting concept drift in industrial data streams

Industrial data streams frequently experience concept drifts. Current drift detection methods, focusing on prediction performance or data distribution, often neglect temporal dependencies and require prior distribution assumptions. These limitations have catalyzed the development of dynamic theory-based approaches that identify drifts by discerning variations in data dynamics. However, the majority of these exhibit high complexity in dynamics characterization and still underperform in industrial drift detection. To overcome these challenges, we propose a singular spectrum analysis-based drift detection method for industry data streams, comprising three modules: common dynamics extraction, noise-robust drift detection, and threshold adaptation. Our approach uses singular spectrum analysis (SSA) as the dynamics representation method instead of traditional high-dimensional phase space embedding and Fourier transform. SSA allows for the efficient description of principal dynamics by intelligently mixing the Fourier basis functions according to given data. The first two modules are specifically engineered to address the common dynamics and stochastic noise inherent in industrial data streams through a well-designed online common dynamics extraction algorithm and a noise-robust detection index respectively. Threshold adaptation module provides a more reasonable way of updating the drift threshold based on the extreme value theory, ensuring global consistency for parameter estimation of extreme value distribution. Our proposed method was tested on simulated examples and a real-world sintering process, exhibiting enhanced drift detection performance.

Tracking and classification performances in the bio-inspired asymmetric and symmetric networks

Abstract Machine learning, deep learning and neural networks are extensively applied for the development of many fields. Though their technologies are improved greatly, they are often said to be opaque in terms of explainability. Their explainable neural functions will be essential to realization in the networks. In this paper, it is shown that the bio-inspired networks are useful for the explanation of tracking and classification of features. First, the asymmetric network with nonlinear functions is created based on the bio-inspired retinal network. They have orthogonal properties useful for the tracking of features compared with the conventional symmetric networks, which is also proposed on the biological functions. Next, the analysis for the independence of the subspaces between the Fourier bases and the asymmetric network bases is performed. It was that the asymmetric networks have better performances in the classification compared with the symmetric ones. Further, the layered asymmetric networks generate the higher dimensional orthogonal bases that improve the classification accuracies by the replacements of bases. Finally, we classified Reuters collections data applying the explainable processing steps, which consist of the linear discriminations and the sparse coding with nearest neighbor relation for classification.

Smoothing algorithms for nonsmooth optimization over the Stiefel manifold with applications to the graph Fourier basis problem

In this paper, we consider a class of nonsmooth and nonconvex optimization problems over the Stiefel manifold where the objective function is the summation of a nonconvex smooth function and a nonsmooth Lipschitz continuous convex function composed with a linear mapping. Besides, we are interested in its application to the graph Fourier basis problem. We propose three numerical algorithms for solving this problem, by combining smoothing methods and some existing algorithms for smooth optimization over the Stiefel manifold. In particular, we approximate the aforementioned nonsmooth convex function by its Moreau envelope in our smoothing methods, and prove that the Moreau envelope has many favorable properties. Thanks to this and the scheme for updating the smoothing parameter, we show that any accumulation point of the solution sequence generated by the proposed algorithms is a stationary point of the original optimization problem. Numerical experiments on building graph Fourier basis are conducted to demonstrate the efficiency of the proposed algorithms.

MetaKernel: Learning Variational Random Features With Limited Labels.

Few-shot learning deals with the fundamental and challenging problem of learning from a few annotated samples, while being able to generalize well on new tasks. The crux of few-shot learning is to extract prior knowledge from related tasks to enable fast adaptation to a new task with a limited amount of data. In this paper, we propose meta-learning kernels with random Fourier features for few-shot learning, we call MetaKernel. Specifically, we propose learning variational random features in a data-driven manner to obtain task-specific kernels by leveraging the shared knowledge provided by related tasks in a meta-learning setting. We treat the random feature basis as the latent variable, which is estimated by variational inference. The shared knowledge from related tasks is incorporated into a context inference of the posterior, which we achieve via a long-short term memory module. To establish more expressive kernels, we deploy conditional normalizing flows based on coupling layers to achieve a richer posterior distribution over random Fourier bases. The resultant kernels are more informative and discriminative, which further improves the few-shot learning. To evaluate our method, we conduct extensive experiments on both few-shot image classification and regression tasks. A thorough ablation study demonstrates that the effectiveness of each introduced component in our method. The benchmark results on fourteen datasets demonstrate MetaKernel consistently delivers at least comparable and often better performance than state-of-the-art alternatives.