Matrix Nearness Problem Research Articles

Matrix nearness problems with off-block-diagonal rank constraints

This paper explores the problem of finding a rank-structured approximation to a given Hermitian positive definite matrix. We first study a fundamental matrix nearness problem which seeks an approximation whose off-diagonal block must have low rank. Our investigation focuses on three objective functions: the spectral norm, Frobenius divergence, and log-determinant divergence. We derive closed-form solutions for the fundamental nearness problem, extending our analysis to include problems with a range restriction on the off-diagonal block. Building upon these results, we propose a practical, scalable algorithm to compute a positive definite hierarchical off-diagonal low-rank approximation to a Hermitian positive definite matrix. Through numerical experiments, we explore the use of our approximation as a general-purpose preconditioner for the conjugate gradient method. Our results demonstrate that the log-determinant divergence-based approximation exhibits good performance across a diverse set of test problems.

Generalized Matrix Nearness Problems

Generalized Matrix Nearness Problems

Quantifying the Structural Stability of Simplicial Homology

Simplicial complexes are generalizations of classical graphs. Their homology groups are widely used to characterize the structure and the topology of data in e.g. chemistry, neuroscience, and transportation networks. In this work we assume we are given a simplicial complex and that we can act on its underlying graph, formed by the set of 1-simplices, and we investigate the stability of its homology with respect to perturbations of the edges in such graph. Precisely, exploiting the isomorphism between the homology groups and the higher-order Laplacian operators, we propose a numerical method to compute the smallest graph perturbation sufficient to change the dimension of the simplex’s Hodge homology. Our approach is based on a matrix nearness problem formulated as a matrix differential equation, which requires an appropriate weighting and normalizing procedure for the boundary operators acting on the Hodge algebra’s homology groups. We develop a bilevel optimization procedure suitable for the formulated matrix nearness problem and illustrate the method’s performance on a variety of synthetic quasi-triangulation datasets and real-world transportation networks.

Algorithm for orthogonal matrix nearness and its application to feature representation

Various learning problems can be represented as certain canonical forms of orthogonal matrix nearness problems under the unitarily invariant norm. Since varying unitarily invariant norms favor different structured learning of input data, it is crucial to construct a unified scheme to learn encouraging distributions via any unitarily invariant norm. In this paper, we find that the orthogonal matrix nearness problem can be generalized to a common architecture of the unitarily invariant norm minimization, thus further exploit iterative and closed-form solutions. Firstly, we start with several special circumstances of orthogonal matrix nearness problems, where the unitarily invariant norm resorts to Frobenius norm. Secondly, a general scheme for orthogonal matrix nearness problems under any unitarily invariant norm is addressed and the corresponding iterative algorithm is proposed. Thirdly, feature representation problems are formulated as certain forms of orthogonal matrix nearness, indicating a joint framework for learning low-dimensional features with closed-form solutions. Finally, comprehensive experiments on real-world data sets demonstrate the effectiveness of the proposed method against compared state-of-the-art feature representation approaches. In particular, the proposed method gains about 15% improvements of clustering accuracy on BASEHOCK, CNAE and RCV1 data sets than benchmarks, and achieves at least 3% improvements on other data sets.

ON ADMM-BASED METHODS FOR SOLVING THE NEARNESS SYMMETRIC SOLUTION OF THE SYSTEM OF MATRIX EQUATIONS <i>A</i><sub>1</sub><i>XB</i><sub>1</sub> = <i>C</i><sub>1</sub> AND <i>A</i><sub>2</sub><i>XB</i><sub>2</sub> = <i>C</i><sub>2</sub>

Two new equivalent forms of the matrix nearness problem are developed. Some sufficient and necessary conditions for a symmetric matrix X? being a solution of the considered problem are presented. Based on the new equivalent forms of the above problem and the idea of the alternating direction method with multipliers (ADMM), we establish two new iterative methods to compute its solution, and analyze the global convergence of the proposed algorithms. Numerical results demonstrate the efficiency of our methods. The development here is an extension of the recent work of Peng, Fang, Xiao and Du [<i>SpringerPlus</i>, 5:1005, 2016] on the nearness symmetric solution of the matrix equation <i>AXB</i> = <i>C</i>.

Measuring the stability of spectral clustering

As an indicator of the stability of spectral clustering of an undirected weighted graph into k clusters, the kth spectral gap of the graph Laplacian is often considered. The kth spectral gap is characterized in this paper as an unstructured distance to ambiguity, namely as the minimal distance of the Laplacian to arbitrary symmetric matrices with vanishing kth spectral gap. As a conceptually more appropriate measure of stability, the structured distance to ambiguity of the k-clustering is introduced as the minimal distance of the Laplacian to Laplacians of graphs with the same vertices and edges but with weights that are perturbed such that the kth spectral gap vanishes. To compute a solution to this matrix nearness problem, a two-level iterative algorithm is proposed that uses a constrained gradient system of matrix differential equations in the inner iteration and a one-dimensional optimization of the perturbation size in the outer iteration. The structured and unstructured distances to ambiguity are compared on some example graphs. The numerical experiments show, in particular, that selecting the number k of clusters according to the criterion of maximal stability can lead to different results for the structured and unstructured stability indicators.

Further results on Moore–Penrose inverses of tensors with application to tensor nearness problems

The notion of the Moore–Penrose inverse of a matrix has been extended to tensor case with Einstein product, recently. In this paper, we continue this work and propose the full rank decomposition of tensors by introducing a new kind of tensor rank, which gives a novel representation of the Moore–Penrose inverse. Moreover, we also obtain several formulae related to the Moore–Penrose inverse of a tensor and its {i,j,k}-inverses. In addition, we consider the tensor nearness problem associated with a tensor equation, a generalization of matrix nearness problems that arise in many areas of applied matrix computations. It is shown that the unique solution to the tensor nearness problem can be represented by using the Moore–Penrose inverses of the given tensors.

Incoherent waveform design for compressed sensing radar based on pulse‐train scenario

Waveform design plays an important role in compressed sensing radar (CSR). The performance of the target scene recovery in CSR scenarios depends on the mutual coherence of the sensing matrix, which is determined by the radar transmission waveform. In this study, the authors propose a waveform optimisation method for pulse-train scenario through solving the matrix nearness problem between the Gram matrix and the identity matrix. This method has a simple closed-form solution and simulation results show that the proposed method is more efficient than the current waveforms such as linear frequency modulation (LFM), Gaussian and Alltop sequences in terms of mutual coherence, recovery errors and recovery percentages.

A Gradient System for Low Rank Matrix Completion

In this article we present and discuss a two step methodology to find the closest low rank completion of a sparse large matrix. Given a large sparse matrix M, the method consists of fixing the rank to r and then looking for the closest rank-r matrix X to M, where the distance is measured in the Frobenius norm. A key element in the solution of this matrix nearness problem consists of the use of a constrained gradient system of matrix differential equations. The obtained results, compared to those obtained by different approaches show that the method has a correct behaviour and is competitive with the ones available in the literature.

Regularization matrices for discrete ill‐posed problems in several space dimensions

SummaryMany applications in science and engineering require the solution of large linear discrete ill‐posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space dimensions. The matrix that defines these problems is very ill conditioned and generally numerically singular, and the right‐hand side, which represents measured data, is typically contaminated by measurement error. Straightforward solution of these problems is generally not meaningful due to severe error propagation. Tikhonov regularization seeks to alleviate this difficulty by replacing the given linear discrete ill‐posed problem by a penalized least‐squares problem, whose solution is less sensitive to the error in the right‐hand side and to roundoff errors introduced during the computations. This paper discusses the construction of penalty terms that are determined by solving a matrix nearness problem. These penalty terms allow partial transformation to standard form of Tikhonov regularization problems that stem from the discretization of integral equations on a cube in several space dimensions.

Erratum to: Centrohermitian and Skew-Centrohermitian Solutions to the Minimum Residual and Matrix Nearness Problems of the Quaternion Matrix Equation $$(AXB,DXE)=(C,F)$$ ( A X B , D X E ) = ( C , F )

Erratum to: Centrohermitian and Skew-Centrohermitian Solutions to the Minimum Residual and Matrix Nearness Problems of the Quaternion Matrix Equation $$(AXB,DXE)=(C,F)$$ ( A X B , D X E ) = ( C , F )

Centrohermitian and Skew-Centrohermitian Solutions to the Minimum Residual and Matrix Nearness Problems of the Quaternion Matrix Equation $${(AXB,DXE) = (C,F)}$$ ( A X B , D X E ) = ( C , F )

It is established that the precise solutions on the minimum residual and matrix nearness problems of the quaternion matrix equation \({(AXB,DXE)=(C,F)}\) for centrohermitian and skew-centrohermitian matrices, where X is an unknown quaternion matrix and A, B, C, D, E, and F are known quaternion matrices of suitable sizes. Moreover, an algorithm to get the solutions of the problems considered is provided, and a numerical example is also given to exemplify the results.

On the matrix nearness problem for (skew-)symmetric matrices associated with the matrix equations (A_1XB_1, ..., A_kXB_k) = (C_1, ..., C_k)

On the matrix nearness problem for (skew-)symmetric matrices associated with the matrix equations (A_1XB_1, ..., A_kXB_k) = (C_1, ..., C_k)

Regularization matrices determined by matrix nearness problems

This paper is concerned with the solution of large-scale linear discrete ill-posed problems with error-contaminated data. Tikhonov regularization is a popular approach to determine meaningful approximate solutions of such problems. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. This matrix should be chosen to promote the recovery of known important features of the desired solution, such as smoothness and monotonicity. We describe a novel approach to determine regularization matrices with desired properties by solving a matrix nearness problem. The constructed regularization matrix is the closest matrix in the Frobenius norm with a prescribed null space to a given matrix. Numerical examples illustrate the performance of the regularization matrices so obtained.

Computing More Consistent Multigroup Nuclear Data Covariances

It is not uncommon that the covariances of multigroup nuclear data do not obey the sum rules of nuclear data. We present a matrix nearness problem of finding a nearest symmetric matrix with given null vectors and solve it when the distance is measured in the Frobenius norm. The problem appears to be new. We propose that the method should be used to find nearest consistent multigroup covariance matrices with respect to the sum rules of redundant nuclear data.If the multigroup covariances cannot be easily interpreted in a consistent manner, there is some ambiguity in choosing values for the covariances that are not explicitly mentioned. We present and compare a simple and a heuristic characterization method.Three practical examples are processed and analyzed: relative covariances of cross sections of 9440Zr and absolute covariances of cross sections of 5024Cr and 23290Th. We demonstrate that satisfactory results can be achieved.We discuss the properties of the proposed method and the characterization methods and suggest possible improvements. The methods can be used as a part of a quality assurance program and might be valuable additions to nuclear data processing codes.

Polynomials and lemniscates of indefiniteness

For a large indefinite linear system, there exists the option to directly precondition for the normal equations. Matrix nearness problems are formulated to assess the attractiveness of this alternative. Polynomial preconditioning leads to polynomial approximation problems involving lemniscate-like sets, both in the plane and in $$\,\mathbb {C}^{n \times n}$$Cn×n. A natural matrix analytic extension for lemniscates is introduced. Operator theoretically one is concerned with polynomial unitarity and associated factorizations for the inverse. For the speed of convergence and lemniscate asymptotics, the notion of quasilemniscate arises. In the $$L^2$$L2-norm algorithms for solving the problem are devised.

Sensing Matrix Restriction Method for Compressed Sensing Radar

To improve the robustness of target scene extraction for compressed sensing radar, an sensing matrix orthogonal restriction method is proposed through solving the matrix nearness problem between the complex Gram matrix and the target matrix. First, we introduce a target matrix maintaining the character of any subset in sensing matrix and minimizing mutual-coherence of sensing matrix. Then least square method and properties of Kronecker product and vector operator are employed to solve the matrix nearness problem with fixed transmission waveform or measurement matrix. The restricted sensing matrix can be achieved through optimizing the transmission waveform and measurement matrix separately or simultaneously. Our approach can effectively reduce the sensing matrix mutual-coherence and improve the performance of target scene recovery. Lastly, numerical simulations are performed to illustrate the priority of the proposed restriction method.

Some matrix nearness problems suggested by Tikhonov regularization

The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for problems of small to moderate size are Tikhonov regularization and truncated singular value decomposition (TSVD). By considering matrix nearness problems related to Tikhonov regularization, several novel regularization methods are derived. These methods share properties with both Tikhonov regularization and TSVD, and can give approximate solutions of higher quality than either one of these methods.

On Matrix Nearness Problems: Distance to Delocalization

In this paper we introduce a new matrix nearness problem that is intended to generalize the distance to instability. Due to its applicability in analyzing the robustness of eigenvalues with respect to the arbitrary localization sets (domains) in the complex plane, we call it the distance to delocalization. For the open left half-plane or the unit disk, the distance to the nearest unstable matrix is obtained as a special case. Following the theoretical framework of Hermitian functions and the Lyapunov-type localization approach, we present new Newton-type algorithms for the distance to delocalization: first using an explicit computation of the desired singular values (eD2D), and then using an implicit computation (iD2D). For both algorithms, we introduce a special stabilization technique of the Newton steps and, for a certain class of the localization domains, we provide an additional globality test. Since our investigations are motivated by several practical applications, we illustrate our approach on som...

Decomposition Methods for Sparse Matrix Nearness Problems

We discuss three types of sparse matrix nearness problems: given a sparse symmetric matrix, find the matrix with the same sparsity pattern that is closest to it in Frobenius norm and (1) is positive semidefinite, (2) has a positive semidefinite completion, or (3) has a Euclidean distance matrix completion. Several proximal splitting and decomposition algorithms for these problems are presented and their performance is compared on a set of test problems. A key feature of the methods is that they involve a series of projections on small dense positive semidefinite or Euclidean distance matrix cones, corresponding to the cliques in a triangulation of the sparsity graph. The methods discussed include the dual block coordinate ascent algorithm (or Dykstra's method), the dual projected gradient and accelerated projected gradient algorithms, and a primal and a dual application of the Douglas--Rachford splitting algorithm.