Hankel Matrix Research Articles

Optimal approximate minimization of one-letter weighted finite automata

Abstract In this paper, we study the approximate minimization problem of weighted finite automata (WFAs): to compute the best possible approximation of a WFA given a bound on the number of states. By reformulating the problem in terms of Hankel matrices, we leverage classical results on the approximation of Hankel operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory. We solve the optimal spectral-norm approximate minimization problem for irredundant WFAs with real weights, defined over a one-letter alphabet. We present a theoretical analysis based on AAK theory and bounds on the quality of the approximation in the spectral norm and $\ell ^2$ norm. Moreover, we provide a closed-form solution, and an algorithm, to compute the optimal approximation of a given size in polynomial time.

Signal Acquisition and Algorithm Design for Bioimpedance-Based Heart Rate Estimation from the Wrist

Background: Heart rate (HR) is a critical biomarker that provides insights into overall health, stress levels, and the autonomic nervous system. Pulse wave signals contain valuable information about the cardiovascular system and heart status. However, signal acquisition in wearables poses challenges, particularly when using electrical sensors, due to factors like the distance from the heart, body movement, and suboptimal electrode placement. Methods: Electrical bioimpedance (EBI) measurements using bipolar and tetrapolar electrode systems were employed for pulse wave signal acquisition from the wrist in both perpendicular and distal configurations. Signal preprocessing techniques, including baseline removal via Hankel matrix methods, normalization, cross-correlation, and peak detection, were applied to improve signal quality. This study describes the combination of sensor-level signal acquisition and processing for accurate wearable HR estimation. Results: The bipolar system was shown to produce larger ΔZ(t), while the tetrapolar system demonstrated higher sensitivity. Distal placement of the electrodes yielded greater ΔZ(t) (up to 0.231 Ω) when targeting both wrist arteries. Bandpass filtering resulted in a better signal-to-noise ratio (SNR), achieving 3.6 dB for the best bipolar setup and 4.8 dB for the tetrapolar setup, compared to 2.6 and 3.3 dB SNR, respectively, with the Savitzky–Golay filter. The custom HR estimation algorithm presented in this paper demonstrated improved accuracy over a reference method, achieving an average error of 1.8 beats per minute for the best bipolar setup, with a mean absolute percentage error (MAPE) of 8%. Conclusions: The analysis supports the feasibility of using bipolar electrode setups on the wrist and highlights the importance of electrode positioning relative to the arteries. The proposed signal processing method, featuring a preprocessing pipeline and HR estimation algorithm, provides a proof-of-concept demonstration for HR estimation from EBI signals acquired at the wrist.

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A direct approach for solving the cubic Szegö equation

AbstractWe study the cubic Szegö equation which is an integrable nonlinear nondispersive and nonlocal evolution equation. In particular, we present a direct approach for obtaining the multiphase and multisoliton solutions as well as a special class of periodic solutions. Our method is substantially different from the existing one which relies mainly on the spectral analysis of the Hankel operator. We show that the cubic Szegö equation can be bilinearized through appropriate dependent variable transformations and then the solutions satisfy a set of bilinear equations. The proof is carried out within the framework of an elementary theory of determinants. Furthermore, we demonstrate that the eigenfunctions associated with the multiphase solutions satisfy the Lax pair for the cubic Szegö equation, providing an alternative proof of the solutions. Last, the eigenvalue problem for a periodic solution is solved exactly to obtain the analytical expressions of the eigenvalues.

Duality, BMO and Hankel operators on Bernstein spaces

In this paper we deal with the problem of describing the dual space (Bκ1)⁎ of the Bernstein space Bκ1, that is the space of entire functions of exponential type (at most) κ>0 whose restriction to the real line is Lebesgue integrable. We provide several characterizations, showing that such dual space can be described as a quotient of the space of entire functions of exponential type κ whose restriction to the real line are in a suitable BMO-type space, or as the space of symbols b for which the Hankel operator Hb is bounded on the Paley–Wiener space Bκ/22. We also provide a characterization of (Bκ1)⁎ as the BMO space w.r.t. the Clark measures of the inner function ei2κz on the upper half-plane, in analogy with the known description of the dual of backward-shift invariant 1-spaces on the torus. Furthermore, we show that the orthogonal projection Pκ:L2(R)→Bκ2 induces a bounded operator from L∞(R) onto (Bκ1)⁎.Finally, we show that Bκ1 is the dual space of a suitable VMO-type space or as the space of symbols b for which the Hankel operator Hb on the Paley–Wiener space Bκ/22 is compact.

Predictive performance of denoising algorithms in S&P 500 and Bitcoin returns

This paper aims to investigate the effectiveness of both the Hankel matrix and the Wavelet denoise methods to address the issue of denoising volatility in datasets, specifically squared returns, and to enhance the accuracy of post-sample forecasting. When converting data from levels to volatility, noise levels are amplified, and the forecasting models suffer from structural bias, leading to reduced accuracy gains. The main challenge in denoising operations is preserving systemic impulses while extracting sporadic and unpredictable components. While conventional denoising algorithms struggle to distinguish systemic patterns from noise, the Hankel matrix approach surpasses them in performance, resulting in improved predictive accuracy. Additionally, this paper explores the Wavelet denoise method, which is found to be less effective than the Hankel matrix approach for GARCH models. The empirical work showcases the denoising process and the accuracy improvements achieved in various datasets for GARCH models. By implementing the Hankel matrix approach, simulations reveal a notable accuracy gain of over 10%.

Ap weights and an application to Hankel operators on Fock spaces with variable exponents on [formula omitted]

We characterize Ap weights for p>1 and via extrapolation we characterize boundedness and compactness of Hankel operators between Fock spaces of variable exponent and the Lebesgue spaces of variable exponents. We also give some characterizations of the symbol class which is some BMO-type spaces with variable exponent on the complex plane Cn.

Forecasting Urban Traffic States with Sparse Data Using Hankel Temporal Matrix Factorization

Forecasting urban traffic states is crucial to transportation network monitoring and management, playing an important role in the decision-making process. Despite the substantial progress that has been made in developing accurate, efficient, and reliable algorithms for traffic forecasting, most existing approaches fail to handle sparsity, high-dimensionality, and nonstationarity in traffic time series and seldom consider the temporal dependence between traffic states. To address these issues, this work presents a Hankel temporal matrix factorization (HTMF) model using the Hankel matrix in the lower dimensional spaces under a matrix factorization framework. In particular, we consider an alternating minimization scheme to optimize the factor matrices in matrix factorization and the Hankel matrix in the lower dimensional spaces simultaneously. To perform traffic state forecasting, we introduce two efficient estimation processes on real-time incremental data, including an online imputation (i.e., reconstruct missing values) and an online forecasting (i.e., estimate future data points). Through extensive experiments on the real-world Uber movement speed data set in Seattle, Washington, we empirically demonstrate the superior forecasting performance of HTMF over several baseline models and highlight the advantages of HTMF for addressing sparsity, nonstationarity, and short time series. History: Accepted by Ram Ramesh, Area Editor for Data Science & Machine Learning. Funding: This research was supported by the Institute for Data Valorisation, the Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation, the National Natural Science Foundation of China [Grants 12371456, 72101049, 72232001], the Sichuan Science and Technology Program [Grant 2024NSFJQ0038], and the Fundamental Research Funds for the Central Universities [Grant DUT23RC(3)045].

Dispersion and Attenuation Estimation of Ultrasonic Guided Waves using Multi-Subarrays Matrix Pencil Method

Abstract Dispersion and attenuation estimation of ultrasonic guided waves is important for waveguide characterization. Matrix pencil (MP) method has been proposed for multimode complex wavenumber estimation, but its performance highly depends on the signal to noise ratio (SNR), which still brings challenges for high attenuation waveguide evaluation. In addition, the model order, i.e., matrix size in MP method, significantly impacts the dispersion curve extraction. In this study, to avoid the model order interference and compensate the SNR sensitivity of the classical MP algorithm, a multi-subarrays MP method is proposed for accurate dispersion and attenuation estimation. Considering with different Hankel matrix sizes, the multi-subarrays MP method estimates the complex wavenumbers from each pair of submatrices to filter uncertain results. A soft threshold strategy is applied to compensate the noise sensitivity of the estimation. Simulated results prove the proposed method can improve the dispersion and attenuation curve estimation.

Essential norm and Schatten class of Hankel operators on doubling Fock spaces

In this paper, we obtain the essential norm of Hankel operators from a doubling Fock space Fφp to a weighted Lebesgue space Lφq for all possible 1≤p,q<∞. We also characterize the Schatten-h class of Hankel operators from Fφ2 to Lφ2.

Denoising of imaginary time response functions with Hankel projections

Imaginary-time response functions of finite-temperature quantum systems are often obtained with methods that exhibit stochastic or systematic errors. Reducing these errors comes at a large computational cost—in quantum Monte Carlo simulations, the reduction of noise by a factor of two incurs a simulation cost of a factor of four. In this paper, we relate certain imaginary-time response functions to an inner product on the space of linear operators on Fock space. We then show that data with noise typically does not respect the positive definiteness of its associated Gramian. The Gramian has the structure of a Hankel matrix. As a method for denoising noisy data, we introduce an alternating projection algorithm that finds the closest positive definite Hankel matrix consistent with noisy data. We test our methodology at the example of fermion Green's functions for continuous-time quantum Monte Carlo data and show remarkable improvements of the error, reducing noise by a factor of up to 20 in practical examples. We argue that Hankel projections should be used whenever finite-temperature imaginary-time data of response functions with errors is analyzed, be it in the context of quantum Monte Carlo, quantum computing, or in approximate semianalytic methodologies. Published by the American Physical Society 2024

Stability analysis of polynomials with an approach of differential topology

SummaryIn this paper, we study the Hurwitz stability of polynomials. By considering that a degree Hurwitz polynomial has its corresponding Markov parameters, we define the set in Section 3, we also define . Based on properties of the Hankel matrices and the stability test, as well as by using ideas of differential topology, we show that is a fiber bundle with a base. This result allows us to obtain an interesting application: Given a Hurwitz polynomial, we can generate two families of positive definite Hankel matrices.

Fast audio signal denoising with low-rank approximation using windowed singular value decomposition of frequency division

The method of constructing Hankel matrices and using low-rank approximation has been demonstrated to be an effective approach to audio denoising. However, the significant computational complexity and the trade-off between the denoising quality and the loss of effective signal remain open issues. This paper proposes a denoising method based on frequency-divided Windowed Singular Value Decomposition (WSVD) and exploits the low-rank characteristics and frequency features commonly found in audio signals including speech and music recordings. The method incorporates an improved Lanczos Bidiagonalization algorithm to accelerate the singular value decomposition with low error and high tolerance. Furthermore, techniques are added at the window junctions to maintain the continuity and smoothness of the final audio, thus achieving denoising efficiently and effectively. This paper also assesses the influence of window segmentation length, main frequency domain characteristics, rank selection of Hankel matrix and characteristics of different noises on the final denoising effect. Finally, the denoising algorithm's robustness and effectiveness are validated through simulations and experiments.

Inverses of Toeplitz plus Hankel operators with generating matrix functions

Inverses of Toeplitz plus Hankel operators with generating matrix functions

Research on rolling bearing fault diagnosis technology based on singular value decomposition

To solve the difficulty of selecting the number of effective singular values in Singular Value Decomposition denoising, a new method to determine the number of effective singular values is proposed. The proposed method to determine the number of effective singular values is based on the non-zero singular value distribution law of the Hankel matrix constructed by the signal. Specifically, the number of effective singular values in the Hankel matrix is twice the number of frequencies contained in the signal, and the difference between the effective singular values of the noisy signal and the non-zero singular values of the pure signal is very small. The proposed method for determining the number of effective singular values is to perform differential processing on the singular values of the signal and normalize the difference obtained. An empirical parameter T is provided, and the number of effective singular values is determined by comparing them with the normalized results. The proposed method is applied to the simulated and measured rolling bearing signals, and the results are compared with the wavelet threshold denoising method. The results show that the proposed method for determining the number of singular values can effectively filter out the noise frequency contained in the signal while maintaining the characteristic frequency of the signal and achieving the purpose of mechanical equipment fault diagnosis.

Blowup algebras of determinantal ideals in prime characteristic

AbstractWe study when blowup algebras are ‐split or strongly ‐regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew‐symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their depth stabilizes. Finally, we show that, for determinantal ideals, there exists a monomial order for which taking initial ideals commutes with taking symbolic powers. To obtain these results, we develop the notion of ‐split filtrations and symbolic ‐split ideals.

Toeplitz operators and Hankel operators on a Bergman space with an exponential weight on the unit ball

Toeplitz operators and Hankel operators on a Bergman space with an exponential weight on the unit ball

Inverses and determinants of three classes of Hankel matrices

Inverses and determinants of three classes of Hankel matrices

Identification of excitation frequencies of a vibrating beam using Dynamic Mode Decomposition

Identifying modal parameters for the different vibrational conditions for any structure or machine is very important for its structural health monitoring. Researchers have proposed different methodologies to determine that structure’s excitation or natural frequency. In the initial design phase, researchers have to rely on building the FE model and then analyze its natural and excitation frequencies. However, for the complex structure, it is difficult to handle a huge dataset of the vibrational information of the mesh for multi-frequency excitations and it becomes computationally expensive to analyze the system’s performance. This work presents the data-driven technique called Dynamic Mode Decomposition (DMD) which addresses this problem by discovering the system’s dynamics using system’s high-dimensional spatiotemporal data. The present work studies the FE vibration of fixed-fixed beams with single and multi-frequency harmonic forcing functions with and without noise for real-time scenarios. Here, DMD is applied to the sequential spatiotemporal data of displacement of the vibrating beams, and dominant frequencies and corresponding mode shapes are identified to predict the actual time dynamics of the system under different conditions in simulation and experimental conditions. This study also involves the failure of the DMD to capture the periodic oscillation (standing wave) in the beam and proposes the application of the Hankel matrix (time-delay coordinate) to capture the actual dynamical parameters of the system. The resulting predicted frequencies match the actual excitation frequencies, and the accuracy is improved with a number of stackings of the given time series data through the Hankel matrix at different rank truncations.

A new method of noise reduction grounded on the Hankel matrix and its application in rubbing fault diagnosis

Abstract In processing signals with singular value decomposition (SVD), one of the keys lies in building an appropriate Hankel matrix from signals. To address the difficulty in extracting the feature information of rubbing faults between rotor and stator, by taking advantage of the nature of rubbing fault information closely related to the rotation period of equipment, a new method of SVD is presented based on the Hankel matrix built from the periodicity of a rotation machine. First, with the periodicity of the rub-impact fault as the basis, the interval step size between Hankel vectors was determined to self-adaptively build a Hankel matrix of signals. Second, the newly-built Hankel matrix was denoised through the singular value differential spectrum. Third, to reduce the loss of data as much as possible, a strategy was proposed to rebuild signals according to the first and last rows of denoised signals. Fourth, features of rubbing faults were extracted according to the frequency spectrum of reconstructed signals, and faults were identified. To verify the applicability and effectiveness of the presented algorithm, various types of simulation signals and tester signals from different states were incorporated. Meanwhile, the presented algorithm was compared with a variety of classical methods. The results prove that the proposed method can not only effectively constrain noise interference, but also highlight fault feature information and correctly identify rub-impact faults.