measurement-based spectral estimation

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 paper addresses high computational complexity in constrained optimization for one-bit/few-bit signal processing—particularly involving semidefinite and low-rank constraints. We introduce the novel concept of “sample-abundant singularity”: when the number of measurements vastly exceeds classical requirements, high-dimensional nonconvex/nonlinear constraints degenerate into an overdetermined linear feasibility problem. Leveraging the finite-volume property, we theoretically establish that this phenomenon guarantees exact signal recovery. Methodologically, we integrate low-precision quantization models with efficient linear feasibility solvers, eliminating iterative optimization and matrix decomposition. Experiments demonstrate that our framework reduces computational cost by several orders of magnitude while maintaining high accuracy in phase retrieval and covariance estimation. Moreover, it exhibits superior practicality in hardware-constrained settings.

This study addresses a fundamental challenge in system identification: distinguishing spurious eigenvalues arising from limited data from those genuinely reflecting the underlying system dynamics. To this end, the paper introduces—for the first time—the probabilistic sampling pseudospectrum \( P(\lambda) \) and its computationally efficient estimator \( \hat{P}(\lambda) \). By leveraging resampling and statistical inference, this framework quantifies the uncertainty of eigenvalues across the complex plane. The proposed approach provides a general and rigorous statistical criterion for data-driven methods such as Dynamic Mode Decomposition and subspace identification, substantially enhancing the reliability of identifying true dynamical modes from noisy, finite-length observations.

Parameter estimation in high-dimensional structured generalized linear models suffers from low efficiency, particularly under realistic design matrices exhibiting anisotropy and strong correlations. Method: This paper introduces a novel spectral estimation framework based on Approximate Message Passing (AMP). Contribution/Results: We provide the first exact asymptotic characterization of spectral estimators under correlated Gaussian designs. We identify a universally optimal covariance-adaptive preprocessing strategy, partially resolving a long-standing conjecture on optimal spectral estimation for rotationally invariant models. Theoretically and empirically, our approach substantially reduces sample complexity and achieves provably statistically optimal estimation accuracy—outperforming existing heuristic methods on canonical designs from computational imaging and genomics.

To address the sequential determination of model order and insufficient robustness in high-dimensional line spectral estimation (LSE), this paper proposes the Bilinear Generalized LSE (BiG-LSE) method. BiG-LSE iteratively approximates the nonlinear observation model via Taylor expansion into a bilinear form and employs Expectation Propagation (EP) for joint Bayesian inference of frequencies, model order, and noise variance. It is the first LSE framework enabling fully automatic order selection and rigorous uncertainty quantification. To enhance sequential estimation accuracy, it adopts the von Mises distribution to model frequency priors. Furthermore, low-rank compression of posterior log-density messages significantly reduces computational complexity. Simulation and real-world experiments demonstrate that BiG-LSE achieves estimation accuracy comparable to state-of-the-art methods while exhibiting superior robustness and practicality under high-dimensional nonlinear measurements.

This study addresses the frequent misinterpretation of fluctuations in dominant eigensubspaces and scalar spectral functionals—such as the absorption ratio—in rolling covariance estimation as genuine market structural changes, when they are often artifacts of estimation noise, particularly under shrinkage. By leveraging perturbation analysis and calibrated inference, the work derives, for the first time, the first-order null distribution of eigensubspace variation under overlapping windows and establishes its invariance under rotation-equivariant shrinkage estimators. It further shows that only scale-invariant spectral functionals enjoy first-order immunity to elliptical kurtosis. To correct high-dimensional bias in the absorption ratio, a trace-preserving spiked debiased estimator is proposed. Theoretical results, supported by Davis–Kahan bounds, distribution-free confidence bands, and an estimator-aware bootstrap, are validated through simulations and successfully applied to equity data for reliable detection of true market structural shifts.

This work addresses the vulnerability of state estimators to unmodeled disturbances—such as sensor aliasing or out-of-distribution noise—and their lack of a general self-assessment mechanism. To this end, the authors propose a sensor-agnostic introspective method that, for the first time, leverages frequency-domain spectral analysis for generic estimator health monitoring. By examining power spectral characteristics—such as signal power, bandwidth, and entropy—of recent velocity estimates, the approach operates without reliance on specific sensor models or assumptions about training data distributions. Experimental results demonstrate that this lightweight method is effective across diverse visual-inertial, LiDAR-inertial, and radar-inertial odometry systems, achieving 51%–58% fault recall and 60%–84% precision on real-world flight datasets, thereby validating the discriminative power of spectral features in detecting estimation failure.

This work addresses the problem of robust mean estimation under the mean-shift contamination model, where an adversary replaces a small fraction of clean samples with arbitrarily shifted samples from the underlying distribution. For general multivariate base distributions whose characteristic functions satisfy mild spectral conditions, the authors propose an efficient Fourier-analytic algorithm capable of achieving mean estimation with arbitrary accuracy. The key innovation lies in the introduction of a “Fourier witness” as a central analytical tool, which enables the first nearly tight characterization of sample complexity for this setting. The derived upper and lower bounds match up to constant factors, thereby revealing the optimal sample efficiency achievable by any robust estimator under this contamination model.

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.