arXiv subjects

Zheng-Jian Bai

Publications and source records attributed to Zheng-Jian Bai.

18 recordsLinked to original sources

arXiv2025

A mixed precision preconditioned Jacobi method for the symmetric eigenvalue problem

The eigenvalue problem is a fundamental problem in scientific computing. In this paper, we first give the error analysis for a single step or sweep of Jacobi's method in floating point arithmetic. Then we propose a mixed precision preconditioned Jacobi method for the symmetric eigenvalue problem: We first compute the eigenvalue decomposition of a real symmetric matrix by an eigensolver at low precision and we obtain a low-precision matrix of eigenvectors; Then by using the high-precision modified Gram-Schmidt orthogonalization process, a high-precision orthogonal matrix is obtained, which is used as an initial guess for Jacobi's method. The rounding error analysis of the proposed method is established under some conditions. We also present a mixed precision preconditioned one-sided Jacobi method for the singular value problem and the corresponding rounding error analysis is discussed. Numerical experiments on CPUs and GPUs are reported to illustrate the efficiency of the proposed method over the original Jacobi method.

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arXiv2024

Two-sided uniformly randomized GSVD for large-scale discrete ill-posed problems with Tikhonov regularizations

The generalized singular value decomposition (GSVD) is a powerful tool for solving discrete ill-posed problems. In this paper, we propose a two-sided uniformly randomized GSVD algorithm for solving the large-scale discrete ill-posed problem with the general Tikhonov regularization. Based on two-sided uniform random sampling, the proposed algorithm can improve the efficiency with less computing time and memory requirement and obtain expected accuracy. The error analysis for the proposed algorithm is also derived. Finally, we report some numerical examples to illustrate the efficiency of the proposed algorithm.

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arXiv2022

A Column-Wise Update Algorithm for Sparse Stochastic Matrix Factorization

Nonnegative matrix factorization arises widely in machine learning and data analysis. In this paper, for a given factorization of rank r, we consider the sparse stochastic matrix factorization (SSMF) of decomposing a prescribed m-by-n stochastic matrix V into a product of an m-by-r stochastic matrix W and an r-by-n stochastic matrix H, where both W and H are required to be sparse. With the prescribed sparsity level, we reformulate the SSMF as an unconstrained nonconvex-nonsmooth minimization problem and introduce a column-wise update algorithm for solving the minimization problem. We show that our algorithm converges globally. The main advantage of our algorithm is that the generated sequence converges to a special critical point of the cost function, which is nearly a global minimizer over each column vector of the W-factor and is a global minimizer over the H-factor as a whole if there is no sparsity requirement on H. Numerical experiments on both synthetic and real data sets are given to demonstrate the effectiveness of our proposed algorithm.

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arXiv2021

A Riemannian Inexact Newton Dogleg Method for Constructing a Symmetric Nonnegative Matrix with Prescribed Spectrum

This paper is concerned with the inverse problem of constructing a symmetric nonnegative matrix from realizable spectrum. We reformulate the inverse problem as an underdetermined nonlinear matrix equation over a Riemannian product manifold. To solve it, we develop a Riemannian underdetermined inexact Newton dogleg method for solving a general underdetermined nonlinear equation defined between Riemannian manifolds and Euclidean spaces. The global and quadratic convergence of the proposed method is established under some mild assumptions. Then we solve the inverse problem by applying the proposed method to its equivalent nonlinear matrix equation and a preconditioner for the perturbed normal Riemannian Newton equation is also constructed. Numerical tests show the efficiency of the proposed method for solving the inverse problem.

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arXiv2021

A Modified Orthogonal Matching Pursuit for Construction of Sparse Probabilistic Boolean Networks

Probabilistic Boolean Networks play a remarkable role in the modelling and control of gene regulatory networks. In this paper, we consider the inverse problem of constructing a sparse probabilistic Boolean network from the prescribed transition probability matrix. We propose a modified orthogonal matching pursuit for solving the inverse problem. We provide some conditions under which the proposed algorithm can recover a sparse probabilistic Boolean network. We also report some numerical results to illustrate the effectiveness of the proposed algorithm.

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arXiv2021

A geometric proximal gradient method for sparse least squares regression with probabilistic simplex constraint

In this paper, we consider the sparse least squares regression problem with probabilistic simplex constraint. Due to the probabilistic simplex constraint, one could not apply the L1 regularization to the considered regression model. To find a sparse solution, we reformulate the least squares regression problem as a nonconvex and nonsmooth L1 regularized minimization problem over the unit sphere. Then we propose a geometric proximal gradient method for solving the regularized problem, where the explicit expression of the global solution to every involved subproblem is obtained. The global convergence of the proposed method is established under some mild assumptions. Some numerical results are reported to illustrate the effectiveness of the proposed algorithm.

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arXiv2020

Single-pass randomized QLP decomposition for low-rank approximation

The QLP decomposition is one of the effective algorithms to approximate singular value decomposition (SVD) in numerical linear algebra. In this paper, we propose some single-pass randomized QLP decomposition algorithms for computing the low-rank matrix approximation. Compared with the deterministic QLP decomposition, the complexity of the proposed algorithms does not increase significantly and the system matrix needs to be accessed only once. Therefore, our algorithms are very suitable for a large matrix stored outside of memory or generated by stream data. In the error analysis, we give the bounds of matrix approximation error and singular value approximation error. Numerical experiments also reported to verify our results.

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arXiv2020

On Pole-Zero Assignment of Vibratory Systems by Multi-Input Feedback Control

In this paper, we consider the pole-zero assignment problem for vibratory systems via multi-input feedback control. We propose a multi-step two-stage approach for solving the multi-input pole-zero assignment problem. We first reformulate the assignment problem as a multi-step single-input pole-zero assignment problem. Then, in each step, we propose a two-stage approach for solving the single-input pole-zero assignment problem. In the first stage, based on the measured receptances, we replace the selected zeros of the prescribed open-loop point receptance to the desired locations, where we need to solve a small underdetermined linear equation for finding the corresponding columns of the feedback matrices. In the second stage, by using linear matrix inequalities, the complete closed-loop poles are assigned to the prescribed subregion of the complex left-hand plane. Finally, we give some numerical examples to demonstrate the effectiveness of our method.

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arXiv2020

Riemannian Newton-CG Methods for Constructing a Positive Doubly Stochastic Matrix From Spectral Data

In this paper, we consider the inverse eigenvalue problem for the positive doubly stochastic matrices, which aims to construct a positive doubly stochastic matrix from the prescribed realizable spectral data. By using the real Schur decomposition, the inverse problem is written as a nonlinear matrix equation on a matrix product manifold. We propose monotone and nonmonotone Riemannian inexact Newton-CG methods for solving the nonlinear matrix equation. The global and quadratic convergence of the proposed methods is established under some assumptions. We also provide invariant subspaces of the constructed solution to the inverse problem based on the computed real Schur decomposition. Finally, we report some numerical tests, including an application in digraph, to illustrate the effectiveness of the proposed methods.

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arXiv2020

Condition numbers for the truncated total least squares problem and their estimations

In this paper, we present explicit expressions for the mixed and componentwise condition numbers of the truncated total least squares (TTLS) solution of $A\boldsymbol{x} \approx \boldsymbol{b} $ under the genericity condition, where $A$ is a $m\times n$ real data matrix and $\boldsymbol{b}$ is a real $m$-vector. Moreover, we reveal that normwise, componentwise and mixed condition numbers for the TTLS problem can recover the previous corresponding counterparts for the total least squares (TLS) problem when the truncated level of for the TTLS problem is $n$. When $A$ is a structured matrix, the structured perturbations for the structured truncated TLS (STTLS) problem are investigated and the corresponding explicit expressions for the structured normwise, componentwise and mixed condition numbers for the STTLS problem are obtained. Furthermore, the relationships between the structured and unstructured normwise, componentwise and mixed condition numbers for the STTLS problem are studied. Based on small sample statistical condition estimation (SCE), reliable condition estimation algorithms for both unstructured and structured normwise, mixed and componentwise are devised, which utilize the SVD of the augmented matrix $[A~\boldsymbol{b} ]$. The efficient proposed condition estimation algorithms can be integrated into the SVD-based direct solver for the small and medium size TTLS problem to give the error estimation for the numerical TTLS solution. Numerical experiments are reported to illustrate the reliability of the proposed estimation algorithms, which coincide with our theoretical results.

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arXiv2020

New Formulation and Computation for Generalized Singular Values of Grassman Matrix Pair

In this paper, we derive new model formulations for computing generalized singular values of a Grassman matrix pair. These new formulations make use of truncated filter matrices to locate the $i$-th generalized singular value of a Grassman matrix pair. The resulting matrix optimization problems can be solved by using numerical methods involving Newton's method on Grassmann manifold. Numerical examples on synthetic data sets and gene expression data sets are reported to demonstrate the high accuracy and the fast computation of the proposed new ormulations for computing arbitrary generalized singular value of Grassman matrix pair.

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arXiv2019

A Riemannian Derivative-Free Polak-Ribiere-Polyak Method for Tangent Vector Field

This paper is concerned with the problem of finding a zero of a tangent vector field on a Riemannian manifold. We first reformulate the problem as an equivalent Riemannian optimization problem. Then we propose a Riemannian derivative-free Polak-Ribiére-Polyak method for solving the Riemannian optimization problem, where a non-monotone line search is employed. The global convergence of the proposed method is established under some mild assumptions. To further improve the efficiency, we also provide a hybrid method, which combines the proposed geometric method with the Riemannian Newton method. Finally, some numerical experiments are reported to illustrate the efficiency of the proposed method.

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arXiv2018

A Structure-Preserving One-Sided Jacobi Method for Computing the SVD of a Quaternion Matrix

In this paper, we provide a structure-preserving one-sided cyclic Jacobi method for computing the singular value decomposition of a quaternion matrix. In this method, the columns of the quaternion matrix are orthogonalized in pairs by using a sequence of orthogonal JRS-symplectic Jacobi matrices to its real counterpart. The quadratic convergence is also established under some mild conditions. Numerical tests are reported to illustrate the efficiency of the proposed method.

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arXiv2018

A Preconditioned Riemannian Gauss-Newton Method for Least Squares Inverse Eigenvalue Problems

This paper is concerned with the least squares inverse eigenvalue problem of reconstructing a linear parameterized real symmetric matrix from the prescribed partial eigenvalues in the sense of least squares, which was originally proposed by Chen and Chu [SIAM J. Numer. Anal., 33 (1996), pp. 2417--2430]. We provide a Riemannian inexact Gausss-Newton method for solving the least squares inverse eigenvalue problem. The global and local convergence analysis of the proposed method is discussed. Also, a preconditioned conjugate gradient method with an efficient preconditioner is proposed for solving the Riemannian Gauss-Newton equation. Finally, some numerical tests, including an application in the inverse Sturm-Liouville problem, are reported to illustrate the efficiency of the proposed method.

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arXiv2018

Perturbation Analysis of An Eigenvector-Dependent Nonlinear Eigenvalue Problem With Applications?

The eigenvector-dependent nonlinear eigenvalue problem (NEPv) $A(P)V=VΛ$, where the columns of $V\in\mathbb{C}^{n\times k}$ are orthonormal, $P=VV^{\mathrm{H}}$, $A(P)$ is Hermitian, and $Λ=V^{\mathrm{H}}A(P)V$, arises in many important applications, such as the discretized Kohn-Sham equation in electronic structure calculations and the trace ratio problem in linear discriminant analysis. In this paper, we perform a perturbation analysis for the NEPv, which gives upper bounds for the distance between the solution to the original NEPv and the solution to the perturbed NEPv. A condition number for the NEPv is introduced, which reveals the factors that affect the sensitivity of the solution. Furthermore, two computable error bounds are given for the NEPv, which can be used to measure the quality of an approximate solution. The theoretical results are validated by numerical experiments for the Kohn-Sham equation and the trace ratio optimization.

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arXiv2017

A Riemannian Inexact Newton-CG Method for Nonnegative Inverse Eigenvalue Problems: Nonsymmetric Case

This paper is concerned with the nonnegative inverse eigenvalue problem of finding a nonnegative matrix such that its spectrum is the prescribed self-conjugate set of complex numbers. We first reformulate the nonnegative inverse eigenvalue problem as an under-determined constrained nonlinear matrix equation over several matrix manifolds. Then we propose a Riemannian inexact Newton-CG method for solving the nonlinear matrix equation. The global and quadratic convergence of the proposed method is established under some mild conditions. We also extend the proposed method to the case of prescribed entries. Finally, numerical experiments are reported to illustrate the efficiency of the proposed method.

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arXiv2013

A Fast Alternating Minimization Algorithm for Total Variation Deblurring Without Boundary Artifacts

Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang [{\em SIAM J. Imaging Sci.}, 1 (2008), pp. 248--272]. The method in a nutshell consists of a discrete Fourier transform-based alternating minimization algorithm with periodic boundary conditions and in which two fast Fourier transforms (FFTs) are required per iteration. In this paper, we propose an alternating minimization algorithm for the continuous version of the total variation image deblurring problem. We establish convergence of the proposed continuous alternating minimization algorithm. The continuous setting is very useful to have a unifying representation of the algorithm, independently of the discrete approximation of the deconvolution problem, in particular concerning the strategies for dealing with boundary artifacts. Indeed, an accurate restoration of blurred and noisy images requires a proper treatment of the boundary. A discrete version of our continuous alternating minimization algorithm is obtained following two different strategies: the imposition of appropriate boundary conditions and the enlargement of the domain. The first one is computationally useful in the case of a symmetric blur, while the second one can be efficiently applied for a nonsymmetric blur. Numerical tests show that our algorithm generates higher quality images in comparable running times with respect to the Fast Total Variation deconvolution algorithm.

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arXiv2010

Fast Preconditioners for Total Variation Deblurring with Anti-Reflective Boundary Conditions

In recent works several authors have proposed the use of precise boundary conditions (BCs) for blurring models and they proved that the resulting choice (Neumann or reflective, anti-reflective) leads to fast algorithms both for deblurring and for detecting the regularization parameters in presence of noise. When considering a symmetric point spread function, the crucial fact is that such BCs are related to fast trigonometric transforms. In this paper we combine the use of precise BCs with the Total Variation (TV) approach in order to preserve the jumps of the given signal (edges of the given image) as much as possible. We consider a classic fixed point method with a preconditioned Krylov method (usually the conjugate gradient method) for the inner iteration. Based on fast trigonometric transforms, we propose some preconditioning strategies which are suitable for reflective and anti-reflective BCs. A theoretical analysis motivates the choice of our preconditioners and an extensive numerical experimentation is reported and critically discussed. The latter shows that the TV regularization with anti-reflective BCs implies not only a reduced analytical error, but also a lower computational cost of the whole restoration procedure over the other BCs.

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