Toeplitz Structure Research Articles

Toeplitz matrix completion via smoothing augmented Lagrange multiplier algorithm

Toplitz matrix completion (TMC) is to fill a low-rank Toeplitz matrix from a small subset of its entries. Based on the augmented Lagrange multiplier (ALM) algorithm for matrix completion, in this paper, we propose a new algorithm for the TMC problem using the smoothing technique of the approximation matrices. The completion matrices generated by the new algorithm are of Toeplitz structure throughout iteration, which save computational cost of the singular value decomposition (SVD) and approximate well the solution. Convergence results of the new algorithm are proved. Finally, the numerical experiments show that the augmented Lagrange multiplier algorithm with smoothing is more effective than the original ALM and the accelerated proximal gradient (APG) algorithms.

Knowledge-Aided Structured Covariance Matrix Estimator Applied for Radar Sensor Signal Detection.

This study deals with the problem of covariance matrix estimation for radar sensor signal detection applications with insufficient secondary data in non-Gaussian clutter. According to the Euclidean mean, the authors combined an available prior covariance matrix with the persymmetric structure covariance estimator, symmetric structure covariance estimator, and Toeplitz structure covariance estimator, respectively, to derive three knowledge-aided structured covariance estimators. At the analysis stage, the authors assess the performance of the proposed estimators in estimation accuracy and detection probability. The analysis is conducted both on the simulated data and real sea clutter data collected by the IPIX radar sensor system. The results show that the knowledge-aided Toeplitz structure covariance estimator (KA-T) has the best performance both in estimation and detection, and the knowledge-aided persymmetric structure covariance estimator (KA-P) has similar performance with the knowledge-aided symmetric structure covariance estimator (KA-S). Moreover, compared with existing knowledge-aided estimator, the proposed estimators can obtain better performance when secondary data are insufficient.

Risk‐adjusted CUSUM charts under model error

In recent years, quality control charts have been increasingly applied in the healthcare environment, for example, to monitor surgical performance. Risk-adjusted cumulative (CUSUM) charts that utilize risk scores like the Parsonnet score to estimate the probability of death of a patient from an operation turn out to be susceptible to misfitted risk models causing deterioration of the charts' properties, in particular, the false alarm behavior. Our approach considers the application of power transformations in the logistic regression model to improve the fit to the binary outcome data. We propose two different approaches of estimating the power exponent δ. The average run length (ARL) to false alarm is calculated with the popular Markov chain approximation in a more efficient way by utilizing the Toeplitz structure of the transition matrix. A sensitivity analysis of the in-control ARL against the true value δ shows potential effects of incorrect choice of δ. Depending on the underlying patient mix, the results vary from robustness to severe impact (doubling of false alarm rate).

A breakdown-free algorithm for computing the determinants of periodic tridiagonal matrices

In this paper, we present a new breakdown-free recursive algorithm for computing the determinants of periodic tridiagonal matrices via a three-term recurrence. Even though the algorithm is not a symbolic algorithm, it never suffers from breakdown. Furthermore, the proposed algorithm theoretically produces exact values for periodic tridiagonal matrices whose entries are all given in integer. In addition, an explicit formula for the determinant of the periodic tridiagonal matrix with Toeplitz structure is also discussed.

Spectrum Sensing Using a Uniform Uncalibrated Linear Antenna Array for Cognitive Radios

In this paper, we study the spectrum sensing problem when the sensing device is equipped with an uncalibrated multiantenna array. We consider an unknown Toeplitz structure for the covariance matrix of the signal of interest and unknown uncorrelated noise variances. We propose two new invariant constant false-alarm rate detectors with less computational cost than the generalized likelihood ratio test: 1) the Rao test and 2) a separating function estimation test (SFET). We calculate approximate expressions for the false-alarm and detection probabilities of the proposed Rao test-based detector. In addition, we derive the exact false-alarm probability of the proposed SFET for the case of two antennas. Moreover, we obtain an accurate approximation for the false-alarm probability of the proposed SFET for the general case using the method of moments. The accuracy of these expressions is validated by Monte-Carlo experiments. The simulation results show that the proposed detectors outperform state-of-the-art detectors.

Spectral Distribution in the Eigenvalues Sequence of Products of g-Toeplitz Structures

Starting from the definition of an $n\times n$ $g$-Toeplitz matrix, $T_{n,g}(u)=\left[\widehat{u}_{r-gs}\right]_{r,s=0}^{n-1},$ where $g$ is a given nonnegative parameter, $\{\widehat{u}_{k}\}$ is the sequence of Fourier coefficients of the Lebesgue integrable function $u$ defined over the domain $\mathbb{T}=(-\pi,\pi]$, we consider the product of $g$-Toeplitz sequences of matrices, $\{T_{n,g}(f_{1})T_{n,g}(f_{2})\},$ which extends the product of Toeplitz structures, $\{T_{n}(f_{1})T_{n}(f_{2})\},$ in the case where the symbols $f_{1},f_{2}\in L^{\infty}(\mathbb{T}).$ Under suitable assumptions, the spectral distribution in the eigenvalues sequence is completely characterized for the products of $g$-Toeplitz structures. Specifically, for $g\geq2$ our result shows that the sequences $\{T_{n,g}(f_{1})T_{n,g}(f_{2})\}$ are clustered to zero. This extends the well-known result, which concerns the classical case (that is, $g=1$) of products of Toeplitz matrices. Finally, a large set of numerical examples confirming the theoretic analysis is presented and discussed.

Sparsity-Inducing DOA Estimation of Coherent Signals Under the Coexistence of Mutual Coupling and Nonuniform Noise

The sparse reconstruction techniques can improve the accuracy and resolution of the direction of arrival (DOA) estimation using sensor arrays. However, due to reflective objects and nonidealities of the antennas and circuitry, the received signals may be coherent and coupled to each other in nonuniform noise environments, causing severe performance degradation of the signal sparse reconstruction. In this paper, a novel sparsity-inducing DOA estimation method is proposed to adapt to such a challenging scenario. To mitigate the nonuniform noise, its power components are first eliminated by a linear transformation. Then, leveraging the steering vector parametrization based on the banded symmetric Toeplitz structure of the mutual coupling matrix (MCM), a reweighted l 1 -norm minimization subject to an error-constrained l 2 -norm is designed to determine the DOA estimates, further enhancing the sparsity and providing robustness against the noise. In addition, a new stochastic Cramer-Rao lower bound (CRLB) of the DOA estimation is derived for the considered adverse condition. The simulation results demonstrate the superiority of the proposed method over its state-of-the-art counterparts.

A fast gradient projection algorithm for time fractional optimal control problem

In this paper a fast gradient projection algorithm for optimal control problems governed by time fractional diffusion equation is developed. The state equation is discretized by piecewise linear FE for space variable and L1 scheme for time variable, while the control variable is approximated by variational discretization. Based on the block triangular Toeplitz structure of the coefficient matrix of the discretized state equation and adjoint state equation, a fast gradient projection algorithm is designed for the control problem. Numerical examples are carried out to illustrate the effectiveness of the algorithms.

Regularizing properties of a class of matrices including the optimal and the superoptimal preconditioners

SummaryIn this work, given a positive definite matrix A, we introduce a class of matrices related to A, which is obtained by suitably combining projections of its powers onto algebras of matrices simultaneously diagonalized by a unitary transform. After a detailed theoretical study of some spectral properties of the matrices of this class, which suggests their use as regularizing preconditioners, we prove that such matrices can be cheaply computed when the matrix A has a Toeplitz structure. We provide numerical evidence of the advantages coming from the employment of the proposed preconditioners when used in regularizing procedures.

An approximate fractional Gaussian noise model with $$\mathcal {O}(n)$$ O ( n ) computational cost

Fractional Gaussian noise (fGn) is a stationary time series model with long-memory properties applied in various fields like econometrics, hydrology and climatology. The computational cost in fitting an fGn model of length n using a likelihood-based approach is $${{\mathcal {O}}}(n^{2})$$ , exploiting the Toeplitz structure of the covariance matrix. In most realistic cases, we do not observe the fGn process directly but only through indirect Gaussian observations, so the Toeplitz structure is easily lost and the computational cost increases to $${{\mathcal {O}}}(n^{3})$$ . This paper presents an approximate fGn model of $${{\mathcal {O}}}(n)$$ computational cost, both with direct and indirect Gaussian observations, with or without conditioning. This is achieved by approximating fGn with a weighted sum of independent first-order autoregressive (AR) processes, fitting the parameters of the approximation to match the autocorrelation function of the fGn model. The resulting approximation is stationary despite being Markov and gives a remarkably accurate fit using only four AR components. Specifically, the given approximate fGn model is incorporated within the class of latent Gaussian models in which Bayesian inference is obtained using the methodology of integrated nested Laplace approximation. The performance of the approximate fGn model is demonstrated in simulations and two real data examples.

Automatic blind deconvolution with Toeplitz-structured sparse total least squares

Given the noise-corrupted seismic recordings, blind deconvolution simultaneously solves for the reflectivity series and the wavelet. Blind deconvolution can be formulated as a fully perturbed linear regression model and solved by the total least-squares (TLS) algorithm. However, this algorithm performs poorly when the data matrix is a structured matrix and ill-conditioned. In blind deconvolution, the data matrix has a Toeplitz structure and is ill-conditioned. Accordingly, we develop a fully automatic single-channel blind-deconvolution algorithm to improve the performance of the TLS method. The proposed algorithm, called Toeplitz-structured sparse TLS, has no assumptions about the phase of the wavelet. However, it assumes that the reflectivity series is sparse. In addition, to reduce the model space and the number of unknowns, the algorithm benefits from the structural constraints on the data matrix. Our algorithm is an alternating minimization method and uses a generalized cross validation function to define the optimum regularization parameter automatically. Because the generalized cross validation function does not require any prior information about the noise level of the data, our approach is suitable for real-world applications. We validate the proposed technique using synthetic examples. In noise-free data, we achieve a near-optimal recovery of the wavelet and the reflectivity series. For noise-corrupted data with a moderate signal-to-noise ratio (S/N), we found that the algorithm successfully accounts for the noise in its model, resulting in a satisfactory performance. However, the results deteriorate as the S/N and the sparsity level of the data are decreased. We also successfully apply the algorithm to real data. The real-data examples come from 2D and 3D data sets of the Teapot Dome seismic survey.

IVDST: A Fast Algorithm for Atomic Norm Minimization in Line Spectral Estimation

This letter presents a fast algorithm named iterative Vandermonde decomposition and shrinkage-thresholding (IVDST), which offers a low-complexity solution to atomic norm minimization (ANM) in off-grid compressed sensing for line spectral estimation from few measurements. It implements the ANM principle via the accelerated proximal gradient (APG) technique, without invoking computationally expensive semidefinite programming (SDP). To approximate the proximal operator in each APG iteration, Vandermonde decomposition is applied to utilize the Toeplitz structure inherent in the line spectral model, and the low-rank property of the Toeplitz-structured matrix is enforced via a simple shrinkage-thresholding operation. The IVDST algorithm effectively reduces the order of computational complexity compared to SDP-based solutions. It also offers an explicit way to bridge the ANM principle with classic super-resolution line spectral estimation algorithms, such as MUSIC.

Efficient DOA Estimation Algorithm for Noncircular Sources Under Unknown Mutual Coupling

Mutual coupling effects may seriously degrade the direction-of-arrival (DOA) estimation performance in practical sensor arrays. In this article, a computationally efficient DOA estimation algorithm is proposed for noncircular sources in the presence of mutual coupling. Utilizing the noncircularity of the sources, we first construct an extended covariance matrix. Then, based on the banded symmetric Toeplitz structure of the mutual coupling matrix, the extended covariance matrix is divided into two submatrices. By simultaneous singular-value decomposition (SVD) of the two submatrices, the DOA parameter is obtained in a closed form. The proposed algorithm can improve the DOA estimation performance. Moreover, it is computationally efficient since it does not require any spectral search and avoids high-order statistics. Simulation results demonstrate both the efficiency and the effectiveness of our proposed algorithm under unknown mutual coupling.

Tensor-Based Angle Estimation Approach for Strictly Noncircular Sources with Unknown Mutual Coupling in Bistatic MIMO Radar

In the paper, the estimation of joint direction-of-departure (DOD) and direction-of-arrival (DOA) for strictly noncircular targets in multiple-input multiple-output (MIMO) radar with unknown mutual coupling is considered, and a tensor-based angle estimation method is proposed. In the proposed method, making use of the banded symmetric Toeplitz structure of the mutual coupling matrix, the influence of the unknown mutual coupling is removed in the tensor domain. Then, a special enhancement tensor is formulated to capture both the noncircularity and inherent multidimensional structure of strictly noncircular signals. After that, the higher-order singular value decomposition (HOSVD) technology is applied for estimating the tensor-based signal subspace. Finally, the direction-of-departure (DOD) and direction-of-arrival (DOA) estimation is obtained by utilizing the rotational invariance technique. Due to the use of both noncircularity and multidimensional structure of the detected signal, the algorithm in this paper has better angle estimation performance than other subspace-based algorithms. The experiment results verify that the method proposed has better angle estimation performance.

Target detection exploiting covariance matrix structures in MIMO radar

Abstract We aim to address the problem of target detection in collocated multiple-input multiple-output (MIMO) radar where the disturbance covariance matrix is unknown. In practice, the disturbance covariance matrix has a persymmetric or Toeplitz structure, when the MIMO radar receiver is equipped with a symmetrically or uniformly spaced linear array. We exploit the persymmetric or Toeplitz structure of the disturbance covariance matrix to design two adaptive detectors that do not require training data. Analytical expressions for the probability of false alarm and detection probability are derived for the proposed detector that takes the persymmetry into account. Numerical examples are provided to show that the proposed detectors outperform the conventional counterparts.

Fast iterative tomographic wavefront estimation with recursive Toeplitz reconstructor structure for large-scale systems.

Tomographic wavefront reconstruction is the main computational bottleneck to realize real-time correction for turbulence-induced wavefront aberrations in future laser-assisted tomographic adaptive-optics (AO) systems for ground-based giant segmented mirror telescopes because of its unprecedented number of degrees of freedom, N, i.e., the number of measurements from wavefront sensors. In this paper, we provide an efficient implementation of the minimum-mean-square error (MMSE) tomographic wavefront reconstruction, which is mainly useful for some classes of AO systems not requiring multi-conjugation, such as laser-tomographic AO, multi-object AO, and ground-layer AO systems, but is also applicable to multi-conjugate AO systems. This work expands that by Conan [Proc. SPIE9148, 91480R (2014)PSISDG0277-786X10.1117/12.2054472] to the multi-wavefront tomographic case using natural and laser guide stars. The new implementation exploits the Toeplitz structure of covariance matrices used in an MMSE reconstructor, which leads to an overall O(N log N) real-time complexity compared with O(N2) of the original implementation using straight vector-matrix multiplication. We show that the Toeplitz-based algorithm leads to 60nm rms wavefront error improvement for the European Extremely Large Telescope laser-tomography AO system over a well-known sparse-based tomographic reconstruction; however, the number of iterations required for suitable performance is still beyond what a real-time system can accommodate to keep up with the time-varying turbulence.

Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the kind $A=T(a)+E$ where $T(a)=(a_{j-i})_{i,j\in \mathbb Z^{+}}$ , $E=(e_{i,j})_{i,j\in \mathbb Z^{+}}$ is compact and the norms $\|a\|_{_{\mathcal {W}}}={\sum }_{i\in \mathbb Z}|a_{i}|$ and $\|E\|_{2}$ are finite. These properties allow to approximate any QT matrix, within any given precision, by means of a finite number of parameters. QT matrices, equipped with the norm $\|A\|_{_{\mathcal {Q}\mathcal {T}}}=\alpha {\|a\|}_{_{\mathcal {W}}}+\|E\|_{2}$ , for $\alpha = (1+\sqrt 5)/2$ , are a Banach algebra with the standard arithmetic operations. We provide an algorithmic description of these operations on the finite parametrization of QT matrices, and we develop a MATLAB toolbox implementing them in a transparent way. The toolbox is then extended to perform arithmetic operations on matrices of finite size that have a Toeplitz plus low-rank structure. This enables the development of algorithms for Toeplitz and quasi-Toeplitz matrices whose cost does not necessarily increase with the dimension of the problem. Some examples of applications to computing matrix functions and to solving matrix equations are presented, and confirm the effectiveness of the approach.

Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix T_n(f), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f. A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where mathbf {f} is an stimes s matrix-valued trigonometric polynomial with sge 1, and T_n(mathbf {f}) is the block Toeplitz matrix generated by mathbf {f}, whose size is N(n,s)=sn. The case s=1 corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrix-less. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the mathbb {Q}_p Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.

Numerical algorithms for mean exit time and escape probability of stochastic systems with asymmetric Lévy motion

For non-Gaussian stochastic dynamical systems, mean exit time and escape probability are important deterministic quantities, which can be obtained from integro-differential (nonlocal) equations. We develop an efficient and convergent numerical method for the mean first exit time and escape probability for stochastic systems with an asymmetric Lévy motion, and analyze the properties of the solutions of the nonlocal equations. The discretized equation has Toeplitz structure that enables utilization of fast Fourier transform in numerical simulations. We also investigate the effects of different system factors on the mean exit time and escape probability, including the skewness parameter, the size of the domain, the drift term and the intensity of Gaussian and non-Gaussian noises. We find that the behavior of the mean exit time and the escape probability has dramatic difference at the boundary of the domain when the index of stability crosses the critical value of one.

A stable explicitly solvable numerical method for the Riesz fractional advection–dispersion equations

In this paper, we present a finite difference scheme for solving the Riesz fractional advection-dispersion equations (RFADEs). The scheme is obtained by using asymmetric discretization technique and modify the shifted Grünwald approximation to fractional derivative. By calculating the unknowns in differential nodal-point sequences at the odd and even time-levels, the discrete solution of the scheme can be obtained explicitly. The computational cost for the scheme at each time step can be O(Klog K) by using the fast matrix-vector multiplication with the help of Toeplitz structure, where K is the number of unknowns. We prove that the scheme is solvable and unconditionally stable. We derive the error estimates in discrete l2-norm, which is optimal in some cases. Numerical examples are presented to verify our theoretical results.