basis function expansion

This study addresses the challenge that traditional threshold time series models struggle to simultaneously capture both abrupt shifts and smooth evolution in threshold parameters. To overcome this limitation, the authors propose a time-varying threshold autoregressive model based on wavelet series expansion, which flexibly approximates irregular jumps and continuous variations in the threshold function. By leveraging the localized time-frequency properties of wavelet bases, the approach circumvents the limitations of Fourier-based methods in modeling local dynamics. Integrating wavelet expansion, a threshold mechanism, and an autoregressive structure, the proposed model demonstrates superior performance in both simulation studies and empirical analyses, achieving significantly higher fitting accuracy and forecasting capability compared to existing methods, thereby offering a novel framework for modeling complex nonlinear time series.

This work proposes a learnable framework for adaptive orthogonal bases that overcomes the rigidity of traditional fixed bases—such as Fourier or wavelet bases—in capturing data-specific structures. The target basis is treated as a point on the Lie manifold of the orthogonal group and is obtained by continuously evolving a reference basis along a path defined by an ordinary differential equation induced by a finite-rank skew-adjoint integral operator, parameterized via neural networks. Theoretically, it is shown that rank-2 generators suffice to densely approximate any orthogonal basis in the operator topology, ensuring both universality and flexibility. Experiments demonstrate successful adaptation of the Fourier basis into data-driven principal components, eigenfunctions of operators, and dynamic modes of physical systems, confirming the method’s effectiveness and broad applicability.

This work addresses the scalability limitations and explicit kernel dependency of traditional kernel methods by proposing a novel paradigm for learning compact, adaptive Hilbert-space representations: an implicit function encoder that learns orthogonal basis functions via neural networks—eliminating the need for predefined kernels. Methodologically, we introduce two training strategies—progressive basis growth and pre-training followed by pruning—and integrate PCA to uncover intrinsic dimensionality. Theoretically, we establish equivalence between neural bases and kernel methods, derive generalization guarantees via Rademacher complexity and PAC-Bayes bounds, and define an interpretable inner-product kernel based on the learned mapping. Experiments on polynomial regression and nonlinear dynamical systems—including the Van der Pol oscillator and two-body orbital dynamics—demonstrate that our approach achieves comparable accuracy with significantly fewer basis functions, offering superior efficiency, interpretability, and kernel-level generalization performance.

Traditional neural networks suffer from limited interpretability and weak theoretical foundations. Method: This paper proposes a novel machine learning paradigm grounded in infinite-dimensional Hilbert spaces, centering on linear operators. It integrates reproducing kernel Hilbert spaces (RKHS), spectral operator learning, wavelet representations, scattering transforms, and Koopman operator theory to formulate learning tasks as sampling, approximation, and dynamical inference in infinite-dimensional function spaces. Contribution/Results: We establish the first unified Hilbert-space-theoretic framework bridging spectral learning and symbolic reasoning. The approach significantly enhances mathematical rigor and model interpretability by grounding learning in well-defined functional-analytic principles. Moreover, it provides a rigorous mathematical foundation and new methodological pathways for deep interdisciplinary integration between signal processing and machine learning—enabling principled analysis of structured data, hierarchical feature extraction, and nonlinear dynamical system modeling.

To address the challenges of basis function selection and modeling observation error correlations in functional data smoothing, this paper proposes a fast Bayesian basis selection method that supports both univariate and multivariate functional data jointly. It is the first to explicitly model the correlation structure of observational errors within a Bayesian framework. The method employs variational inference combined with an EM algorithm, replacing conventional MCMC sampling to achieve substantial computational speedup without compromising estimation accuracy. Extensive evaluations on simulated data and real-world applications—including motorcycle acceleration and Canadian weather datasets—demonstrate significant improvements in adjusted R², accurate recovery of basis coefficients, optimal basis sets, and within-curve correlation patterns. An open-source R implementation is publicly available.

This work addresses the limitations of conventional spectral neural operators, which rely on fixed global bases and struggle to capture spatial heterogeneity and multiscale dynamics. The authors propose the Adaptive Basis Learning (ABLE) framework—the first approach to enable end-to-end learning of spectral bases. ABLE constructs data-driven, spatially adaptive Parseval frames that preserve invertibility and maintain O(N log N) computational complexity, effectively shifting representational capacity from spectral coefficients to the basis functions themselves. The framework leverages an FFT-based efficient implementation, incorporates learnable auxiliary density functions, and can seamlessly replace spectral layers in existing neural operators. Experiments demonstrate that ABLE significantly outperforms strong baselines across multiple PDE benchmarks, particularly excelling in scenarios with sharp gradients and multiscale features. Moreover, when integrated as a plug-in module into models such as U-FNO and HPM, it consistently enhances performance.

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.

Existing state space models based on polynomial bases struggle to effectively capture nonstationary signals with localized or transient structures. This work proposes a multiscale state space model grounded in wavelet frames, which for the first time incorporates the local support property of wavelets into state space modeling, replacing conventional global polynomial bases. This substitution substantially enhances the model’s capacity to represent localized temporal dynamics. By integrating the continuous-time projection framework of HiPPO, the approach achieves both efficiency and stability in long-sequence modeling. Empirical evaluations on real-world datasets featuring transient structures—such as PTB-XL electrocardiograms and raw audio from Speech Commands—demonstrate superior performance over orthogonal-basis methods like S4.

This study addresses the challenge of constructing neurally plausible, coherent continuous-space representations that unify physical and perceptual information. Building upon the spatial semantic pointer framework, we propose a radial basis kernel implementation tailored for distributed representations and systematically analyze the role of grid-cell-like codes within this architecture. Both theoretical analysis and empirical results demonstrate that grid-cell-like representations not only naturally facilitate the construction of radial basis kernels but also exhibit optimality in terms of neural implementability. Our work is the first to integrate spatial semantic pointers, distributed Fourier embeddings, and grid-cell-like mechanisms into a unified model, yielding an efficient and biologically plausible spatial representation system that demonstrates significant advantages in representational capacity and neural implementation efficiency.

This work addresses the challenge that traditional time-invariant models struggle to effectively capture switching dynamics in time-varying systems. To overcome this limitation, the authors propose a neural network–based time-varying state-space model that incorporates a learnable dictionary of time-varying basis functions. This design flexibly represents diverse temporal evolution patterns of system dynamics while maintaining manageable computational complexity, substantially enhancing the model’s capacity to capture switching sequences. The study further reveals an optimal allocation strategy for time-varying degrees of freedom across model components. Experimental results demonstrate that the proposed model consistently outperforms existing time-invariant approaches on both synthetic switching systems and speech denoising tasks, confirming its effectiveness and strong generalization capability.