spectral perturbation analysis

In strong interference scenarios, modal components and interference become coupled in time-frequency spectrograms of multicomponent signals, severely degrading ridge detection accuracy. Method: This paper proposes a spectrogram decoupling framework that pioneers the integration of texture–geometry decomposition into time-frequency analysis. It establishes a dual-path architecture comprising a variational optimization model and a U-Net-based supervised learning network to separate intrinsic mode components from interference components in spectrograms. Furthermore, it introduces an interference-estimation-driven local adaptive window-length selection criterion, overcoming the limitations of fixed window lengths. Results: Evaluated on a synthetic multi-interference spectrogram dataset, the method significantly improves ridge localization accuracy (average gain of +27%), demonstrating both high precision and strong robustness. The two complementary pathways synergistically enhance overall performance.

This work addresses the limitation of existing output-confidence–based fault detection methods, which often fail to capture internal errors in neural networks. The authors propose Self-Detecting Neural Networks (SDNN), a novel framework that introduces the concept of “spectral drift” to reveal that erroneous predictions manifest as pronounced multi-scale spectral instabilities in internal activations. Spectral features are extracted via short-time Fourier transform, wavelet decomposition, and statistical moments, and a lightweight detector is trained using curriculum learning to establish an end-to-end learnable internal monitoring mechanism. Evaluated on CIFAR-10, SDNN achieves an AUROC of 79.0 ± 25.3%, outperforming baseline methods such as MaxSoftmax and Energy Score by 25–30 percentage points.

This paper addresses the quantile spectrum and cross-spectrum—nonstationary frequency-domain features defined over the two-dimensional domain of frequency and quantile level—originally proposed by Li (2012, 2014), and introduces the first nonparametric estimation framework for them. Methodologically, it constructs the quantile discrete Fourier transform (QDFT) and quantile spectral sequences (QSER) via trigonometric quantile regression; spectral estimators are then built from the autocovariance function of QSER using windowing techniques, augmented by novel inter-quantile smoothing to enhance estimation stability. The main contributions are: (i) establishing the first rigorous theoretical framework for QDFT–QSER, enabling fully nonparametric modeling of quantile spectra; and (ii) delivering an estimator that, in simulations, achieves superior statistical accuracy and robustness compared to the classical L-W estimator—particularly through substantial variance reduction—thereby providing a generalizable tool for quantile-based spectral analysis.

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 investigates the impact of matrix concatenation operations on singular value spectra, aiming to bridge a theoretical gap in the structural stability of SVDs under concatenation. Addressing the central question—“How are the singular values of a concatenated matrix determined by those of its constituent submatrices?”—we extend Weyl’s inequality to block-wise concatenation for the first time, establishing a quantitative analytical framework grounded in matrix perturbation theory and norm inequalities. We derive computable upper bounds on singular value deviations and rigorously prove that dominant singular values remain stable when the operator norms of the submatrices are comparable. This result provides theoretical guarantees and principled design guidance for low-rank approximation, robust matrix clustering, and compression algorithms.

This work addresses the challenge of training instability in Transformers, which often leads to divergence and wasted computational resources without reliable early warning. The authors introduce Koopman spectral analysis into Transformer stability research for the first time, proposing a method that extracts dynamic modal features from inter-layer residual snapshots via a single forward pass at initialization. This enables the construction of a “near-identity spectral quality” metric that predicts divergence risk with high accuracy (AUROC = 0.995). Furthermore, they design a Koopman Spectral Shaping (KSS) mechanism to actively regulate the spectral distribution during training, thereby enhancing stability. The approach reduces the divergence rate from 66.7% to 12.5% under aggressive settings—without normalization layers and with high learning rates—and permits learning rate increases of 50%–150%.

This work addresses the lack of theoretical characterization in existing kernel methods for machine learning regarding the residual structure and energy stability of multichannel signals in complex systems. The authors propose an analytical framework grounded in operator defect identities, introducing the novel concept of “telescopic energy residuals.” By integrating iterative products with a λₙ-relaxed Kaczmarz scheme, they establish admissibility conditions for residuals and derive prior energy bounds. For the first time, this framework incorporates operator defect theory into kernel methods and kernel principal component analysis (KPCA), rigorously proving explicit convergence of generalized algorithms, a residual energy decomposition theorem, and stability criteria under noise. The approach significantly extends infinite-dimensional Kaczmarz theory to broader applications in machine learning.

This study addresses the problem of quantifying the goodness-of-fit of moving average MA(q) models to the spectral density of stationary processes by proposing a spectral-domain coefficient of determination. This coefficient extends, for the first time, the classical notion of the coefficient of determination into the framework of spectral analysis to measure how closely an MA(q) model approximates the true spectral density. Constructed via periodogram-based estimation, the proposed coefficient is shown to possess asymptotic normality under rigorous derivation, enabling the development of both a model order selection criterion and a goodness-of-fit test specifically tailored for MA(q) models. The approach adaptively identifies the minimal order q that achieves a pre-specified accuracy level, offering a method that is theoretically sound and practically useful.

Fluorescence background and noise severely impede critical feature identification in Raman spectroscopy. To address this, we propose a Fourier-domain fractional-order variational filtering method: it embeds fractional-order derivatives into a frequency-domain variational model, establishing an optimization framework that jointly achieves denoising and spectral feature preservation; Shannon entropy is introduced to adaptively tune both the regularization parameter and the differentiation order, ensuring accurate retention of chemically meaningful features—including peak positions, intensities, and integrated areas. By synergizing the structural expressiveness of variational modeling with the computational efficiency of frequency-domain processing, our method effectively suppresses fluorescence and noise in both synthetic and real-world Raman spectra while preserving subtle spectral details. Experiments demonstrate its high robustness, superior computational efficiency, and fully automated parameter selection—eliminating manual tuning. This work establishes an interpretable, deployable paradigm for Raman spectral preprocessing.

This study addresses the challenge of effectively identifying high-frequency dynamic signals in electroencephalography (EEG) that are associated with neurological disorders. For the first time, dynamic mode decomposition (DMD) is applied to analyze high-frequency EEG components. By extracting stable and consistent high-frequency dynamic features from neurologically relevant channels, the authors construct discriminative feature representations through a pipeline integrating principal component analysis (PCA), statistical testing based on random distribution assumptions, and machine learning–based classification. Experimental results reveal significant and stable high-frequency dynamic patterns in approximately 70% of the samples, which successfully differentiate individuals with alcohol dependence from healthy controls, thereby demonstrating the method’s effectiveness and novelty.