trend-amplitude decomposition

This work addresses the interpretable additive decomposition of one-dimensional signals. We propose the first Transformer-based end-to-end deep learning method that automatically disentangles an input signal into four physically meaningful components: piecewise-constant, smooth (low-frequency), texture (high-frequency), and noise. Unlike conventional approaches relying on hand-crafted mathematical priors (e.g., total variation or sinusoidal models), our method employs a sequence-to-sequence architecture with a multi-branch output head to jointly predict all components, trained exclusively on synthetically generated data. Experiments on in-distribution synthetic signals demonstrate significantly lower reconstruction errors for each component compared to classical methods—including TV-L1 regularization and synchrosqueezing transform—validating the Transformer’s capacity to learn effective signal priors from data. The results establish a new data-driven paradigm for interpretable signal decomposition, highlighting both modeling efficacy and generalization potential.

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.

Blaschke decomposition (PDU) suffers from modeling difficulties for complex trends and amplitude modulations in nonstationary signals, as well as mode mixing induced by phase wrapping. To address these issues, this paper proposes a divide-and-conquer strategy incorporating windowed segmentation and cumulative-sum (cumsum) preprocessing: the signal is partitioned via a sliding window, tapered windows suppress boundary effects, and cumsum integration prior to phase unwrapping provides smooth, unwrapped-phase initialization—thereby mitigating phase discontinuities and enhancing multiresolution decomposition stability. The method significantly improves adaptability to strongly nonstationary signals. Extensive validation on both synthetic and real-world data demonstrates superior decomposition accuracy and accelerated convergence compared to conventional approaches. The framework maintains theoretical rigor—grounded in Hardy space theory—while offering practical utility for engineering applications in time-frequency analysis and signal separation.

This work addresses heteroscedastic stochastic time series by proposing a dual-signal decomposition framework that disentangles the original series into three components: mean, dispersion (i.e., time-varying volatility), and stationary white noise. Methodologically, it employs a dual-output neural network—or equivalently, a nonlinear optimization model—to jointly model the dynamics of mean and dispersion. An adaptive regularization weighting mechanism, grounded in statistical process control, is introduced to balance learning objectives. Crucially, first- and second-order temporal derivative regularizers are integrated to enforce signal smoothness, suppress noise, and preserve structural integrity. Two complementary learning paradigms—joint learning and sequential learning—are developed to enable collaborative modeling, cross-effect analysis, and multi-series structural comparison in a 2D signal space. Experiments demonstrate accurate separation of uncorrelated noise, robust handling of abrupt and smooth regime shifts, substantial improvements in predictive accuracy and interpretability for both signals, and strong generalization and extensibility across diverse applications.

This paper addresses the weak theoretical foundations of matrix decomposition in machine learning by systematically constructing a self-consistent, comprehensive, and modern-application-oriented pedagogical framework. Methodologically, it grounds the exposition in numerical linear algebra and matrix analysis, unifying classical decompositions—including LU, QR, SVD, and block triangular factorizations—while integrating numerical stability analysis and Hermitian/Hilbert space theory. Crucially, it bridges traditional numerical analysis with deep learning’s backpropagation setting, emphasizing differentiability and computational robustness of decompositions in algorithm design and model optimization. The primary contribution is a compact, dual-purpose (teaching and research) knowledge system that fills critical gaps in both theoretical coherence and machine-learning relevance present in existing literature, thereby providing rigorous mathematical foundations for high-dimensional data modeling and efficient training.

This study addresses the trade-off between computational efficiency and physiological interpretability in the analysis of oscillatory biomedical signals such as ECG and EEG. It establishes, for the first time, a rigorous mathematical equivalence between finite-order adaptive Fourier decomposition (AFD) and the frequency modulation model (FMM) based on Möbius transformations, further demonstrating that both frameworks correspond to the same optimization problem under a Gaussian noise assumption. By unifying these approaches, the method integrates Takenaka–Malmquist orthogonal bases, parametric modeling, and FFT-based acceleration, and naturally extends to multichannel settings. Experimental results show that this equivalence effectively combines the computational efficiency of AFD with the physiological interpretability of FMM, leading to significantly improved signal decomposition performance.

This work addresses the efficient approximation of the Koopman operator for nonlinear dynamical systems by proposing a novel construction of observables based on the continuous wavelet transform. It establishes, for the first time, a rigorous proof that these wavelet-based observables serve as eigenfunctions of the Koopman semigroup in a specific Banach space, and derives closed-form expressions for both the operator’s action and its resolvent. Building upon this theoretical foundation, the authors integrate the approach with Extended Dynamic Mode Decomposition (EDMD) to formulate a new algorithmic framework, termed cWDMD. Numerical experiments demonstrate that the proposed method achieves high-accuracy approximations of the Koopman operator, significantly enhancing both the precision and computational efficiency of spectral analysis for nonlinear systems.

This study systematically investigates the frequency-domain encoding capabilities of the Chronos foundation model, addressing a critical gap in understanding how such models represent fundamental signal properties. Through controlled experiments using discrete sinusoidal signals and a lightweight online Minimum Description Length (MDL) probing framework, the work examines the existence, separability, and cross-spectral fidelity of internal frequency representations within the Chronos decoder. The research reveals, for the first time, a degradation in representation quality in high-frequency regions, thereby delineating both the strengths and limitations of Chronos’s frequency encoding mechanism. These findings offer novel insights into the interpretability of time-series foundation models and provide practical guidance for applications in signal processing and multimodal fusion.

This work addresses the challenge that existing time series forecasting models struggle to achieve interpretable decomposition of multiple effects—such as trend and seasonality—using time-domain smoothing methods. The authors propose MLOW, an interpretable decomposition framework grounded in the frequency domain’s amplitude spectrum, which captures dominant effects through low-rank representations. To mitigate spectral leakage, MLOW incorporates a flexible mechanism for input windowing and frequency selection. Its core innovation is Hyperplane Non-negative Matrix Factorization (Hyperplane-NMF), which balances interpretability with computational efficiency and generalization. MLOW enables hierarchical, noise-robust multi-effect decomposition and can be seamlessly integrated as a plug-and-play module into mainstream forecasting architectures. Experimental results across multiple benchmarks demonstrate significant performance gains with minimal modifications, while visualizations confirm the clarity and validity of its decompositions.