fourier transform

Existing diffusion models rely heavily on heuristic, empirically designed noise schedules lacking theoretical grounding, leading to spectral discrepancies between generated and real data distributions. This work introduces the first spectral-analysis-based framework for noise schedule design: it formalizes diffusion sampling as a closed-form frequency-domain transfer function, enabling theoretically principled alignment between the noise schedule and the intrinsic spectral characteristics of data. Under assumptions of Gaussianity and translation invariance, we derive an analytical frequency-domain response model and develop a data-driven, adaptive schedule optimization algorithm. The resulting frequency-aware noise schedules significantly improve sampling efficiency—accelerating generation by up to 1.8×—and enhance sample quality, reducing FID by 12.3%. Our approach establishes a mathematically interpretable foundation for noise scheduling and provides a practical, spectrum-informed design paradigm for diffusion models.

Nonparametric spectral density estimation for continuous-time processes observed under irregular spatiotemporal sampling remains challenging due to aliasing and the severe ill-posedness of conventional nonuniform Fourier inversion. Method: This paper proposes the Weighted Nonuniform Fourier Sum (WUNFS) estimator, which introduces a high-accuracy adaptive window function to construct a nonuniform Fourier quadrature framework—thereby suppressing aliasing at its source and circumventing the ill-conditioning inherent in standard nonuniform Fourier inversion. Contribution/Results: We derive a theoretical bias bound for WUNFS and demonstrate its natural extensibility to multivariate settings. Experiments show that WUNFS significantly outperforms both the periodogram and the Lomb–Scargle periodogram (LSP), especially for spectra with slow decay and under multidimensional irregular sampling, achieving substantial gains in estimation accuracy and robustness.

This study addresses the limitations of conventional image enhancement, filtering, and pattern recognition—namely, heavy reliance on manual feature engineering and insufficient real-time performance—by proposing a theory-driven, end-to-end machine learning framework. Methodologically, it is the first to systematically integrate discrete Fourier transform (DFT), Z-transform, and continuous Fourier analysis into deep learning pipelines, synergistically coupling them with convolutional neural networks (CNNs) and classical digital filtering algorithms to enable frequency-domain-guided automated feature extraction and real-time joint signal–image processing. The key contributions include: (i) development of an extensible Python framework; (ii) average PSNR improvement of 3.2 dB in image enhancement and noise suppression tasks; and (iii) 40% acceleration in feature extraction efficiency. This work establishes a novel paradigm for AI-powered real-time computer vision that simultaneously ensures high performance and interpretability.

Diffusion models exhibit a Fourier-domain inductive bias in the standard DDPM forward process: high-frequency components suffer excessively rapid SNR decay, causing the reverse process to violate the Gaussianity assumption and degrading high-frequency detail synthesis. To address this, we propose an isochronous frequency-domain noising scheme, wherein all frequency components are progressively corrupted at identical rates—eliminating the inherent frequency-dependent hierarchy in conventional diffusion. This design fundamentally redefines diffusion dynamics modeling from the Fourier perspective, requiring no modifications to the reverse sampler. Experiments demonstrate substantial improvements in generation quality for high-frequency–dominant tasks sensitive to edges and textures, while maintaining performance parity with DDPM on standard image benchmarks including CIFAR-10 and CelebA.

Traditional discrete Fourier transform (DFT) is constrained by uniform sampling and fixed-length sequences, rendering it inadequate for non-uniformly sampled, missing-data, or ultra-long signals. To address this, we propose the Extended Discrete Fourier Transform (EDFT), which formulates spectral estimation as an optimization problem minimizing the Fourier integral residual. EDFT adaptively constructs frequency-domain basis functions without requiring equispaced time-domain sampling or identical sequence lengths. Our method integrates iterative optimization, explicit Fourier integral constraints, and adaptive inverse DFT-based signal reconstruction. It enables high-resolution spectral estimation, time-domain extrapolation, missing-data imputation, and direct processing of non-uniformly sampled signals. Compared to DFT, EDFT substantially broadens the applicability of Fourier analysis while preserving theoretical rigor and computational feasibility.

This work proposes a novel method for predicting the generalization performance of deep neural networks without requiring a validation set. By analyzing the frequency-domain characteristics of parameter trajectories during SGD optimization, the study introduces Fourier fractal dimension as a new generalization metric and establishes a theoretical framework based on the characteristic function of Lévy-driven stochastic differential equations. Building upon this insight, the authors design a trainable frequency-domain regularization mechanism together with a tailored optimizer to actively control the fractal dimension and thereby enhance generalization. Experiments on CIFAR-10, SVHN, and MNIST demonstrate that the proposed approach significantly outperforms existing generalization measures in terms of Kendall rank correlation coefficient, offering both accurate prediction and effective optimization capabilities.

This work addresses the limitation of existing lossy compression methods, which typically prioritize spatial-domain accuracy while failing to preserve spectral fidelity—critical for scientific applications such as cosmology and turbulent combustion. The authors propose a fast Fourier correction algorithm that, for the first time, establishes a joint spatial–spectral error constraint model built upon outputs from compressors like SZ3 and ZFP. By iteratively projecting reconstruction errors into the intersection of allowable error bounds in both domains and leveraging GPU acceleration, the method achieves efficient dual-domain fidelity preservation. Experiments demonstrate that the approach significantly retains essential spatial structures and spectral characteristics across diverse datasets, including cosmological simulations, X-ray diffraction data, combustion models, and electroencephalography signals, thereby balancing high accuracy with practical usability.

Existing diffusion models typically employ pointwise reconstruction losses that are insensitive to signal spectra and multiscale structures, often yielding samples with imbalanced frequency content and insufficient structural detail. To address this limitation, this work proposes a lightweight, plug-and-play spectral regularization framework that introduces differentiable Fourier- and wavelet-domain losses as soft inductive biases during standard training. The approach requires no modifications to the diffusion process, model architecture, or sampling procedure, and is compatible with mainstream paradigms such as DDPM, DDIM, and EDM. Evaluated on high-resolution unconditional generation tasks for both images and audio, the method consistently enhances sample quality—particularly in terms of spectral fidelity and multiscale coherence—while incurring negligible computational overhead.

This study addresses the performance degradation of hyperspectral image classification under single-source domain generalization, primarily caused by domain shift due to sensor discrepancies and high-dimensional spectral characteristics. To mitigate this issue, the authors propose a spectral attribute-driven adaptive data augmentation approach that enhances model generalization to unseen target domains. The method introduces a spectral diversity module and an adaptive channel mixer through spectral dimension resampling and interpolation between adjacent spectral bands, coupled with a spatial-spectral co-optimization mechanism to simultaneously preserve sample fidelity and enrich spectral diversity. Extensive experiments on three benchmark hyperspectral remote sensing datasets demonstrate that the proposed approach significantly outperforms current state-of-the-art methods, effectively improving cross-domain generalization capability.

This work addresses the challenges of extending diffusion models to spherical data, where conventional Euclidean approaches fail due to geometric and stochastic complexities. By operating in a finite-dimensional frequency domain spanned by spherical harmonics, the authors formulate a diffusion framework via the spherical discrete Fourier transform, mapping spatial Brownian motion to a constrained Gaussian process with deterministic, anisotropic covariance. They establish forward and reverse stochastic differential equations tailored to the sphere and rigorously derive the associated frequency-domain diffusion dynamics and noise covariance structure. Crucially, the study reveals the inequivalence between spatial- and spectral-domain score-matching objectives on the sphere, introducing geometry-aware inductive biases. This represents the first diffusion-based generative framework defined directly in the spherical harmonic coefficient space, offering a principled foundation and novel methodology for generating spherical signals in domains such as climate modeling and astronomy.