Low-rank Hankel Matrix Research Articles

Reconstruction of GPS Coordinate Time Series Based on Low-Rank Hankel Matrix Recovery

Reconstruction of GPS Coordinate Time Series Based on Low-Rank Hankel Matrix Recovery

Exponential Signal Reconstruction With Deep Hankel Matrix Factorization.

Exponential function is a basic form of temporal signals, and how to fast acquire this signal is one of the fundamental problems and frontiers in signal processing. To achieve this goal, partial data may be acquired but result in severe artifacts in its spectrum, which is the Fourier transform of exponentials. Thus, reliable spectrum reconstruction is highly expected in the fast data acquisition in many applications, such as chemistry, biology, and medical imaging. In this work, we propose a deep learning method whose neural network structure is designed by imitating the iterative process in the model-based state-of-the-art exponentials' reconstruction method with the low-rank Hankel matrix factorization. With the experiments on synthetic data and realistic biological magnetic resonance signals, we demonstrate that the new method yields much lower reconstruction errors and preserves the low-intensity signals much better than compared methods.

Pattern synthesis for symmetric linear transducer array by bilinear factorization of Hankel matrices

We investigate the problem of designing an unequally spaced symmetric linear array with a minimum number of elements. For unequally spaced arrays, simultaneously tuning all the elements' positions and weights leads to a non-convex problem. Inspired by the fact that the matrix pencil method (MPM) can estimate the weights and positions of array elements from a Hankel matrix formed by samples of beampattern, we attack the design problem by finding the Hankel matrix fulfilling specifications with the minimum rank. Fortunately, the low-rank optimization can be addressed by nuclear norm relaxation. In the proposed approach, we minimize the nuclear norm of the Hankel matrix utilizing the low-rank matrix factorization-based augmented Lagrangian alternating direction method. Once the low-rank Hankel matrix is obtained, we finish the array design with the help of MPM. By finding the desired Hankel matrix with a minimum nuclear norm, the proposed approach avoids the NP-hard problem of simultaneously optimizing the weights and positions of elements. Moreover, the proposed method does not need a reference beampattern and makes no assumptions on element positions, making it more flexible in facing real-world problems. We compare the proposed approach with existing methods under different scenarios and prove its effectiveness.

Structured Gradient Descent for Fast Robust Low-Rank Hankel Matrix Completion

We study the robust matrix completion problem for the low-rank Hankel matrix, which detects the sparse corruptions caused by extreme outliers while we try to recover the original Hankel matrix from partial observation. In this paper, we explore the convenient Hankel structure and propose a novel nonconvex algorithm, coined Hankel structured gradient descent (HSGD), for large-scale robust Hankel matrix completion problems. HSGD is highly computing- and sample-efficient compared to the state of the art. The recovery guarantee with a linear convergence rate has been established for HSGD under some mild assumptions. The empirical advantages of HSGD are verified on both synthetic datasets and real-world nuclear magnetic resonance signals.

Non-uniformly sampled 2D NMR Spectroscopy reconstruction based on Low Rank Hankel Matrix Fast Tri-Factorization and Non-convex Factorization

Nuclear magnetic resonance (NMR) spectroscopy is widely used in chemistry and medicine to research the composition of matter and the spatial structure of proteins. However, NMR signals with larger sizes and higher dimensions are slower to acquire. Therefore, Non-Uniform Sampling (NUS) methods are used to speed up the acquisition of NMR signals, and some mathematical algorithms are used to recover the original NMR signals from the NUS data. In recent years, low-rank Hankel matrix completion is proved to recover NMR data with less error. However, it requires singular value decomposition (SVD) with high computational complexity in each iteration, which is not suitable for reconstructing signals with large sizes. In this paper, we propose a Fast Tri-Factorization (FTF) method to decompose the low-rank Hankel matrix into three small-scale matrices, and it convert the original problem into a matrix nuclear norm minimization problem on the small-scale matrix to mitigate high computation cost of multiple SVDs. Furthermore, the factor matrix size remains constant for different size of the Hankel matrices. We further replace the low-rank constraint with Non-convex Factorization (NF) to avoid the time-consuming SVD in each iteration. Numerical experiments show that the proposed method can significantly speed up the NMR reconstruction compared with several existing algorithms while ensuring the reconstruction accuracy.

Fast NMR spectroscopy reconstruction with a sliding window based Hankel matrix

Nuclear Magnetic Resonance (NMR) spectroscopy is one of the most promising analytical chemistry techniques, although it takes a long time to acquire data. Non-uniform sampling (NUS) is an effective way to reduce the sampling time, but faithful reconstruction methods are needed. The low rank Hankel matrix (LRHM) approach uses the low rank constraint to obtain high-quality spectra from NUS signals, but the reconstruction has a considerable time overhead. In this work, we propose a sliding window based low rank Hankel matrix approach to speed up the spectra reconstruction from NUS signals. Using the sliding window to construct a matrix can effectively reduce the size of the Hankel matrix for faster reconstructions. To further decrease the reconstruction time, parallel computation is applied in the proposed approach. The experiments on both synthetic data and realistic data demonstrate that the reconstruction speed of the proposed method is the fastest among compared methods without sacrificing the quality of spectra.

A penalized method of alternating projections for weighted low-rank hankel matrix optimization

Weighted low-rank Hankel matrix optimization has long been used to reconstruct contaminated signal or forecast missing values for time series of a wide class. The Method of Alternating Projections (MAP) (i.e., alternatively projecting to a low-rank matrix manifold and the Hankel matrix subspace) is a leading method. Despite its wide use, MAP has long been criticized of lacking convergence and of ignoring the weights used to reflect importance of the observed data. The most of known results are in a local sense. In particular, the latest research shows that MAP may converge at a linear rate provided that the initial point is close enough to a true solution and a transversality condition is satisfied. In this paper, we propose a globalized variant of MAP through a penalty approach. The proposed method inherits the favourable local properties of MAP and has the same computational complexity. Moreover, it is capable of handling a general weight matrix, is globally convergent, and enjoys local linear convergence rate provided that the cutting off singular values are significantly smaller than the kept ones. Furthermore, the new method also applies to complex data. Extensive numerical experiments demonstrate the efficiency of the proposed method against several popular variants of MAP.

Bayesian robust hankel matrix completion with uncertainty modeling for synchrophasor data recovery

Synchrophasor data suffer from quality issues like missing and bad data. Exploiting the low-rankness of the Hankel matrix of the synchrophasor data, this paper formulates the data recovery problem as a robust low-rank Hankel matrix completion problem and proposes a Bayesian data recovery method that estimates the posterior distribution of synchrophasor data from partial observations. In contrast to the deterministic approaches, our proposed Bayesian method provides an uncertainty index to evaluate the confidence of each estimation. To the best of our knowledge, this is the first method that provides confidence measure for synchrophasor data recovery. Numerical experiments on synthetic data and recorded synchrophasor data demonstrate that our method outperforms existing low-rank matrix completion methods.

Image restoration: Structured low rank matrix framework for piecewise smooth functions and beyond

Recently, mapping a signal/image into a low rank Hankel/Toeplitz matrix has become an emerging alternative to the traditional sparse regularization, due to its ability to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this paper, we introduce a novel structured low rank matrix framework to restore piecewise smooth functions. Inspired by the total generalized variation to use sparse higher order derivatives, we derive that the Fourier samples of higher order derivatives satisfy an annihilation relation, resulting in a low rank multi-fold Hankel matrix. We further observe that the SVD of a low rank Hankel matrix corresponds to a tight wavelet frame system which can represent the image with sparse coefficients. Based on this observation, we also propose a wavelet frame analysis approach based continuous domain regularization model for the piecewise smooth image restoration. Finally, numerical results on image restoration tasks are presented as a proof-of-concept study to demonstrate that the proposed approach is compared favorably against several popular discrete regularization approaches and structured low rank matrix approaches.

A fast low rank Vandermonde factorization reconstruction method for non-uniformly sampled 2D NMR spectroscopy

As an indispensable analytical method in the fields of chemistry and medicine, nuclear magnetic resonance (NMR) spectroscopy is widely adopted for analyzing the composition and structure of substances. However, the data acquisition period for high-resolution NMR spectra is extremely long. Therefore, non-uniform sampling is commonly adopted to reduce the measurement time and the complete NMR signal can be obtained by appropriate reconstruction algorithms. Recently, it was found that NMR signal can be reconstructed well by low-rank Hankel matrix completion. However, the computational time required for this method to reconstruct two-dimensional (2D) and higher dimensional signals is relatively long because it increases rapidly as spectral resolution improves. In this paper, we propose a novel low rank minimization method to alleviate this problem by utilizing the unexploited property that a 2D NMR signal can be decomposed into two Vandermonde matrices. However, minimizing the rank necessitates the time-consuming singular value decomposition (SVD). Thus we introduce a matrix bi-factorization idea into the method to reduce the time complexity and greatly mitigate the time required by the SVD. Experiments on synthetic data and realistic NMR spectroscopy data demonstrate that compared with the state-of-the-art nuclear norm based minimization and fast iterative hard thresholding algorithms, the computational time of the proposed algorithm is significantly reduced, and high reconstruction accuracy can be ensured at the same time.

A fast Hankel matrix nonconvex factorization reconstruction method with inertia momentum for non-uniformly sampled NMR spectroscopy

Multidimensional nuclear magnetic resonance (NMR) spectroscopy is one of the most powerful tools for qualitative or quantitative analysis of the composition and structure of various organic and inorganic substances. However, the time required to acquire NMR signals increases exponentially with dimensionality. Therefore, non-uniform sampling is commonly adopted to accelerate data acquisition and then the complete spectrum can be obtained by reconstruction method. At present, the state-of-the-art reconstruction methods are based on the idea of low-rank Hankel matrix completion and solved by different singular value thresholding methods. However, the computation of singular value decomposition (SVD) is very time-consuming, especially for high-resolution spectra. In this paper, we proposed a Hankel matrix nonconvex factorization optimization model to avoid SVD, thus greatly reducing the computational time. We developed a numerical algorithm based on the linearized Alternating direction method of multipliers to solve the proposed optimization problem, and Nesterov momentum is adopted to accelerate the convergence of the algorithm. However, for nonconvex model, the high quality reconstruction can only be obtained when the number of the spectral peaks is known. Thus we designed an adaptive strategy to alleviate this problem. Experiments are performed to demonstrate that the proposed algorithm can obtain higher quality reconstruction in less computational time than low rank Hankel matrix factorization and fast iterative hard thresholding methods.

Fast Cadzow’s Algorithm and a Gradient Variant

The Cadzow’s algorithm is a signal denoising and recovery method which was designed for signals corresponding to low rank Hankel matrices. In this paper we first introduce a Fast Cadzow’s algorithm which is developed by incorporating a novel subspace projection to reduce the high computational cost of the SVD in the Cadzow’s algorithm. Then a Gradient method and a Fast Gradient method are proposed to address the non-decreasing MSE issue when applying the Cadzow’s or Fast Cadzow’s algorithm for signal denoising. Extensive empirical performance comparisons demonstrate that the proposed algorithms can complete the denoising and recovery tasks more efficiently and effectively.

Spectrally Sparse Signal Recovery via Hankel Matrix Completion With Prior Information

This paper studies the problem of reconstructing spectrally sparse signals from a small random subset of time domain samples via low-rank Hankel matrix completion with the aid of prior information. By leveraging the low-rank structure of spectrally sparse signals in the lifting domain and the similarity between the signals and their prior information, we propose a convex method to recover the undersampled spectrally sparse signals. The proposed approach integrates the inner product of the desired signal and its prior information in the lift domain into vanilla Hankel matrix completion, which maximizes the correlation between the signals and their prior information. Theoretical analysis indicates that when the prior information is reliable, the proposed method has a better performance than vanilla Hankel matrix completion, which reduces the number of measurements by a logarithmic factor. We also develop an ADMM algorithm to solve the corresponding optimization problem. Numerical results are provided to verify the performance of proposed method and corresponding algorithm.

On Low-Rank Hankel Matrix Denoising

The low-complexity assumption in linear systems can often be expressed as rank deficiency in data matrices with generalized Hankel structure. This makes it possible to denoise the data by estimating the underlying structured low-rank matrix. However, standard low-rank approximation approaches are not guaranteed to perform well in estimating the noise-free matrix. In this paper, recent results in matrix denoising by singular value shrinkage are reviewed. A novel approach is proposed to solve the low-rank Hankel matrix denoising problem by using an iterative algorithm in structured low-rank approximation modified with data-driven singular value shrinkage. It is shown numerically in both the input-output trajectory denoising and the impulse response denoising problems, that the proposed method performs the best in terms of estimating the noise-free matrix among existing algorithms of low-rank matrix approximation and denoising.

An Automatic Denoising Method for NMR Spectroscopy Based on Low-Rank Hankel Model

Nuclear magnetic resonance (NMR) spectroscopy, whose time domain data is modeled as the sum of damped exponential signals, has become an indispensable tool in various scenarios, such as biomedicine, biology, and chemistry. NMR spectroscopy signals, however, are usually corrupted by Gaussian noise in practice, raising difficulties in sequential analysis and quantification. The low-rank Hankel property of exponential signals plays an important role in the denoising issue, but selecting an appropriate parameter still remains a problem. In this work, we explore the effect of the regularization parameter of a convex optimization denoising method based on low-rank Hankel matrices for exponential signals corrupted by Gaussian noise. An accurate estimate on the spectral norm of weighted Hankel matrices is provided as a guidance to set the regularization parameter. The bound can be efficiently calculated since it only depends on the standard deviation of the noise and a constant. Aided by the bound, one can easily obtain an auto-setting regularization parameter to produce promising denoised results. Our results on synthetic and realistic NMR spectroscopy data demonstrate a superior denoising performance of the proposed approach over typical Cadzow and the state-of-the-art QR decomposition methods, especially in the low signal-to-noise ratio regime.

Review and prospect: NMR spectroscopy denoising and reconstruction with low-rank Hankel matrices and tensors.

Nuclear magnetic resonance (NMR) spectroscopy is an important analytical tool in chemistry, biology, and life science, but it suffers from relatively low sensitivity and long acquisition time. Thus, improving the apparent signal-to-noise ratio and accelerating data acquisition became indispensable. In this review, we summarize the recent progress on low-rank Hankel matrix and tensor methods, which exploit the exponential property of free-induction decay signals, to enable effective denoising and spectra reconstruction. We also outline future developments that are likely to make NMR spectroscopy a far more powerful technique.

Accelerated Structured Alternating Projections for Robust Spectrally Sparse Signal Recovery

Consider a spectrally sparse signal $\boldsymbol{x}$ that consists of $r$ complex sinusoids with or without damping. We study the robust recovery problem for the spectrally sparse signal under the fully observed setting, which is about recovering $\boldsymbol{x}$ and a sparse corruption vector $\boldsymbol{s}$ from their sum $\boldsymbol{z}=\boldsymbol{x}+\boldsymbol{s}$. In this paper, we exploit the low-rank property of the Hankel matrix formed by $\boldsymbol{x}$, and formulate the problem as the robust recovery of a corrupted low-rank Hankel matrix. We develop a highly efficient non-convex algorithm, coined Accelerated Structured Alternating Projections (ASAP). The high computational efficiency and low space complexity of ASAP are achieved by fast computations involving structured matrices, and a subspace projection method for accelerated low-rank approximation. Theoretical recovery guarantee with a linear convergence rate has been established for ASAP, under some mild assumptions on $\boldsymbol{x}$ and $\boldsymbol{s}$. Empirical performance comparisons on both synthetic and real-world data confirm the advantages of ASAP, in terms of computational efficiency and robustness aspects.

A Fast Self-Learning Subspace Reconstruction Method for Non-Uniformly Sampled Nuclear Magnetic Resonance Spectroscopy

Multidimensional nuclear magnetic resonance (NMR) spectroscopy is one of the most crucial detection tools for molecular structure analysis and has been widely used in biomedicine and chemistry. However, the development of NMR spectroscopy is hampered by long data collection time. Non-uniform sampling empowers rapid signal acquisition by collecting a small subset of data. Since the sampling rate is lower than that of the Nyquist sampling ratio, undersampling artifacts arise in reconstructed spectra. To obtain a high-quality spectrum, it is necessary to apply reasonable prior constraints in spectrum reconstruction models. The self-learning subspace method has been shown to possess superior advantages than that of the state-of-the-art low-rank Hankel matrix method when adopting high acceleration in data sampling. However, the self-learning subspace method is time-consuming due to the singular value decomposition in iterations. In this paper, we propose a fast self-learning subspace method to enable fast and high-quality reconstructions. Aided by parallel computing, the experiment results show that the proposed method can reconstruct high-fidelity spectra but spend less than 10% of the time required by the non-parallel self-learning subspace method.

Data Driven Tight Frame for Compressed Sensing MRI Reconstruction via Off-the-Grid Regularization

Recently, the finite-rate-of-innovation (FRI) based continuous domain regularization is emerging as an alternative to the conventional on-the-grid sparse regularization for the compressed sensing (CS) due to its ability to alleviate the basis mismatch between the true support of the shape in the continuous domain and the discrete grid. In this paper, we propose a new off-the-grid regularization for the CS-MRI reconstruction. Following the recent works on two dimensional FRI, we assume that the discontinuities/edges of the image are localized in the zero level set of a band-limited periodic function. This assumption induces the linear dependencies among the Fourier samples of the gradient of the image, which leads to a low rank two-fold Hankel matrix. We further observe that the singular value decomposition of a low rank Hankel matrix corresponds to an adaptive tight frame system which can represent the image with sparse canonical coefficients. Based on this observation, we propose a data driven tight frame based off-the-grid regularization model for the CS-MRI reconstruction. To solve the nonconvex and nonsmooth model, a proximal alternating minimization algorithm with a guaranteed global convergence is adopted. Finally, the numerical experiments show that our proposed data driven tight frame based approach outperforms the existing approaches.

Compressive Multi-Attribute Data Gathering Using Hankel Matrix in Wireless Sensor Networks

For heterogeneous WSNs with various types of sensors, compressive data gathering method requires more measurements due to the increased multiple attributes. In this letter, a compressive multi-attribute data gathering method using low-rank Hankel matrix is proposed to reduce the required measurements and improve the recovery accuracy in heterogeneous WSNs. Beyond utilizing just the spatiotemporal correlation of the raw sensed data with compressed sensing, the proposed method further enforces the low-rank block Hankel matrix to exploit the inherent correlation among multi-attribute data. Experimental results demonstrate that the proposed method can significantly improve the recovery accuracy of multi-attribute data compared with the existing solutions in WSNs.