Componentwise Condition Numbers Research Articles

On condition numbers of quaternion matrix inverse and quaternion linear systems with multiple right-hand sides

This paper is devoted to the condition numbers of quaternion linear system with multiple right-hand sides and the associated condition numbers of the quaternion matrix inverse as well. The explicit expressions of the unstructured and structured normwise, mixed, and componentwise condition numbers for the system are given. To reduce the computational cost of the condition numbers,compact and tight upper bounds for these condition numbers are proposed. For general sparse and badly scaled problems, numerical examples show that mixed and componentwise condition numbers are preferred than the normwise condition number for estimating the forward error of the solution, and structured condition numbers are tighter than the unstructured ones for some specific structured problems.

Adversarial ink: componentwise backward error attacks on deep learning

Abstract Deep neural networks are capable of state-of-the-art performance in many classification tasks. However, they are known to be vulnerable to adversarial attacks—small perturbations to the input that lead to a change in classification. We address this issue from the perspective of backward error and condition number, concepts that have proved useful in numerical analysis. To do this, we build on the work of Beuzeville, T., Boudier, P., Buttari, A., Gratton, S., Mary, T. and Pralet S. (2021) Adversarial attacks via backward error analysis. hal-03296180, version 3. In particular, we develop a new class of attack algorithms that use componentwise relative perturbations. Such attacks are highly relevant in the case of handwritten documents or printed texts where, for example, the classification of signatures, postcodes, dates or numerical quantities may be altered by changing only the ink consistency and not the background. This makes the perturbed images look natural to the naked eye. Such ‘adversarial ink’ attacks therefore reveal a weakness that can have a serious impact on safety and security. We illustrate the new attacks on real data and contrast them with existing algorithms. We also study the use of a componentwise condition number to quantify vulnerability.

Condition numbers of the mixed least squares-total least squares problem revisited

A recent study on the condition numbers of the mixed least squares-total least squares (MTLS) problem is due to Zheng and Yang (Numer Linear Algebra Appl. 2019;26(4):e2239). However, the associated expressions are not compact and the Kronecker-product operations make the computation costly. In this paper, we first present new and alternative closed formula for the first order perturbation estimate and condition numbers of the MTLS solution. Then we reveal the relationship between the new formula and Zheng and Yang's result. Several new computable formulae and perturbation bounds for the normwise condition number of the MTLS solution are also provided. Finally, mixed and componentwise condition numbers, structured condition numbers are investigated. Through a number of tests, they are shown to be tighter than the normwise condition numbers for sparse and structured problems.

Perturbation analysis and condition numbers for the Tikhonov regularization of total least squares problem and their statistical estimation

Condition number plays an important role in perturbation analysis, the latter is a tool to judge whether a numerical solution makes sense, especially for ill-posed problems. In this paper, perturbation analysis of the Tikhonov regularization of total least squares problem (TRTLS) is considered. The explicit expressions of normwise, mixed and componentwise condition numbers for the TRTLS problem are first presented. With the intermediate result, i.e. normwise condition number, we can recover the upper bound of TRTLS problem. To improve the computational efficiency in calculating the normwise condition number, a new compact and tight upper bound of the TRTLS problem is introduced. In addition, we also derive the normwise, mixed and componentwise condition numbers for TRTLS problem when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed. We choose the probabilistic spectral norm estimator and the small-sample statistical condition estimation method to estimate these condition numbers with high reliability. Numerical experiments are provided to verify the obtained results.

On condition numbers of the total least squares problem with linear equality constraint

This paper is devoted to condition numbers of the total least squares problem with linear equality constraint (TLSE). With novel limit techniques, closed formulae for normwise, mixed and componentwise condition numbers of the TLSE problem are derived. Computable expressions and upper bounds for these condition numbers are also given to avoid the costly Kronecker product-based operations. The results unify the ones for the TLS problem. For TLSE problems with equilibratory input data, numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, mixed and componentwise condition numbers-based estimates are much tighter.

ARIC: accurate and robust inference of cell type proportions from bulk gene expression or DNA methylation data.

Quantifying cell proportions, especially for rare cell types in some scenarios, is of great value in tracking signals associated with certain phenotypes or diseases. Although some methods have been proposed to infer cell proportions from multicomponent bulk data, they are substantially less effective for estimating the proportions of rare cell types which are highly sensitive to feature outliers and collinearity. Here we proposed a new deconvolution algorithm named ARIC to estimate cell type proportions from gene expression or DNA methylation data. ARIC employs a novel two-step marker selection strategy, including collinear feature elimination based on the component-wise condition number and adaptive removal of outlier markers. This strategy can systematically obtain effective markers for weighted $\upsilon$-support vector regression to ensure a robust and precise rare proportion prediction. We showed that ARIC can accurately estimate fractions in both DNA methylation and gene expression data from different experiments. We further applied ARIC to the survival prediction of ovarian cancer and the condition monitoring of chronic kidney disease, and the results demonstrate the high accuracy and robustness as well as clinical potentials of ARIC. Taken together, ARIC is a promising tool to solve the deconvolution problem of bulk data where rare components are of vital importance.

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

AbstractIn this paper, we present explicit expressions for the mixed and componentwise condition numbers of the truncated total least squares (TTLS) solution of under the genericity condition, where A is a m × n real data matrix and 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 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. We devise reliable condition estimation algorithms for the TTLS problem by utilizing small‐sample statistical condition estimation techniques. The proposed condition estimation algorithms employ the singular value decomposition (SVD) of the augmented matrix to reduce the computational complexity, where both unstructured and structured normwise, mixed, and componentwise condition estimations are considered. The 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 condition estimation algorithms.

Condition numbers of the minimum norm least squares solution for the least squares problem involving Kronecker products

<abstract> In this paper, the upper bounds for two kinds of normiwse condition numbers are derived for $ \min\limits_{x}\|(A\otimes B)x-b\|_2 $ when the coefficient matrix is of rank deficient. In addition, the upper bounds on the mixed and componentwise condition numbers are also given. Numerical experiments are given to confirm our results. </abstract>

Perturbation Analysis for the Matrix-Scaled Total Least Squares Problem

In this paper, we extend matrix scaled total least squares (MSTLS) problem with a single right-hand side to the case of multiple right-hand sides. Firstly, under some mild conditions, this paper gives an explicit expression of the minimum norm solution of MSTLS problem with multiple right-hand sides. Then, we present the Kronecker-product-based formulae for the normwise, mixed and componentwise condition numbers of the MSTLS problem. For easy estimation, we also exhibit Kronecker-product-free upper bounds for these condition numbers. All these results can reduce to those of the total least squares (TLS) problem which were given by Zheng et al. Finally, two numerical experiments are performed to illustrate our results.

Condition Numbers for a Linear Function of the Solution to the Constrained and Weighted Least Squares Problem and Their Statistical Estimation

In this paper, we consider the condition number theory for a linear function of the solution to the constrained and weighted least squares problem. We first present two explicit expressions without Kronecker product of normwise condition number using the classical method for condition numbers. Then, we derive the explicit expression of mixed and componentwise condition numbers by the dual techniques. To estimate these condition numbers with high reliability, we choose the probabilistic spectral norm estimator and the small-sample statistical condition estimation method and devise three algorithms. Numerical experiments are provided to illustrate the obtained results.

Structured condition numbers and statistical condition estimation for the LDU factorization

In this article, we consider the structured condition numbers for LDU, factorization by using the modified matrix-vector approach and the differential calculus, which can be represented by sets of parameters. By setting the specific norms and weight parameters, we present the expressions of the structured normwise, mixed, componentwise condition numbers and the corresponding results for unstructured ones. In addition, we investigate the statistical estimation of condition numbers of LDU factorization using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, we compare the structured condition numbers with the corresponding unstructured ones in numerical experiments.

The spectral properties of Vandermonde matrices with clustered nodes

We study rectangular Vandermonde matrices V with N+1 rows and s irregularly spaced nodes on the unit circle, in cases where some of the nodes are “clustered” together – the elements inside each cluster being separated by at most h≲1N, and the clusters being separated from each other by at least θ≳1N. We show that any pair of column subspaces corresponding to two different clusters are nearly orthogonal: the minimal principal angle between them is at mostπ2−c1Nθ−c2Nh, for some constants c1,c2 depending only on the multiplicities of the clusters. As a result, spectral analysis of VN is significantly simplified by reducing the problem to the analysis of each cluster individually. Consequently we derive accurate estimates for 1) all the singular values of V, and 2) componentwise condition numbers for the linear least squares problem. Importantly, these estimates are exponential only in the local cluster multiplicities, while changing at most linearly with s.

Fourth-order tensor Riccati equations with the Einstein product

ABSTRACT In this paper, the fourth-order tensor Riccati equation with the Einstein product is investigated. Using the Einstein product, we introduce the -tensor, the spectrum and the block tensor multiplication. We also present an expression of the fourth-order permutation tensors and generalize the Riccati equation from the matrix to the tensor case. The existence and uniqueness of the solution for the fourth-order -tensor Riccati equation are studied. Furthermore, we provide a perturbation analysis of the mixed and componentwise condition numbers for the -tensor Riccati equation. An algorithm for solving the equation is proposed. Numerical examples are presented to show the effectiveness of our algorithm.

Condition numbers of the multidimensional total least squares problems having more than one solution

Recently, the condition numbers of the total least squares (TLS) problems having a unique solution have been studied at length in Zheng et al. (SIAM J. Matrix Anal. Appl. 38: 924–948, 2017). However, it is known that the TLS problem may have no solution, and even if an existing solution, it may not be unique. As a continuation of their work, in this paper, we investigate the condition numbers of the minimum Frobenius norm solution of the (multidimensional) TLS problem when having more than one solution. The tight and computable upper bound estimates of the normwise, mixed, and componentwise condition numbers are respectively derived. Some numerical experiments are performed to illustrate the tightness of these upper bounds.

Sensitivity analysis for the generalized Cholesky factorization

The explicit expressions of the normwise, mixed, and componentwise condition numbers and their upper bounds for the generalized Cholesky factorization are first obtained. Then, some improved rigorous perturbation bounds with normwise or componentwise perturbation in the given matrix are derived by bringing together the modified matrix-vector equation approach with the method of Lyapunov majorant function and the Banach fixed point theorem. Theoretical and experimental results show that these new bounds are always tighter than the corresponding ones in the literature.

Condition numbers for the K-weighted pseudoinverse and their statistical estimation

The explicit expressions of normwise, mixed and componentwise condition numbers for the K-weighted pseudoinverse are first presented. With the intermediate result, i.e. the derivative of , we can recover the explicit expressions of condition numbers for solution of least squares problem with equality constraint. Then, we investigate the statistical estimation of condition numbers of using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, numerical examples are provided to illustrate these results.

Condition numbers of generalized saddle point systems

In this paper, we consider the normwise, mixed and componentwise condition numbers of generalized saddle point system. The explicit expressions and the easily computable upper bounds for these condition numbers are presented. Numerical experiments are given to illustrate the effectiveness of the proposed upper bounds.

Perturbation analysis for mixed least squares–total least squares problems

SummaryIn many linear parameter estimation problems, one can use the mixed least squares–total least squares (MTLS) approach to solve them. This paper is devoted to the perturbation analysis of the MTLS problem. Firstly, we present the normwise, mixed, and componentwise condition numbers of the MTLS problem, and find that the normwise, mixed, and componentwise condition numbers of the TLS problem and the LS problem are unified in the ones of the MTLS problem. In the analysis of the first‐order perturbation, we first provide an upper bound based on the normwise condition number. In order to overcome the problems encountered in calculating the normwise condition number, we give an upper bound for computing more effectively for the MTLS problem. As two estimation techniques for solving the linear parameter estimation problems, interesting connections between their solutions, their residuals for the MTLS problem, and the LS problem are compared. Finally, some numerical experiments are performed to illustrate our results.

Condition Numbers of the Nonlinear Matrix Equation Xp-A⁎eXA=I

We explore the condition numbers of the nonlinear matrix equation Xp-A⁎eXA=I. Explicit expressions for the normwise, mixed, and componentwise condition numbers are derived. The upper bounds for the mixed and componentwise condition numbers are obtained. The numerical result favors the fact that our estimations are fairly sharp. Also, the relative upper perturbation bounds give satisfactory results for small perturbations in the input data.

Condition numbers for a linear function of the solution of the linear least squares problem with equality constraints

In this paper, we consider the normwise, mixed and componentwise condition numbers for a linear function L x of the solution x to the linear least squares problem with equality constraints (LSE). The explicit expressions of the normwise, mixed and componentwise condition numbers are derived. Also, we revisit some previous results on the condition numbers of linear least squares problem (LS) and LSE. It is shown that some previous explicit condition number expressions on LS and LSE can be recovered from our new derived condition numbers’ formulas. The sharp upper bounds for the derived normwise, mixed and componentwise condition numbers are obtained, which can be estimated efficiently by means of the classical Hager–Higham algorithm for estimating matrix one-norm. Moreover, the proposed condition estimation methods can be incorporated into the generalized QR factorization method for solving LSE. The numerical examples show that when the coefficient matrices of LSE are sparse and badly-scaled, the mixed and componentwise condition numbers can give sharp perturbation bounds, on the other hand normwise condition numbers can severely overestimate the exact relative errors because normwise condition numbers ignore the data sparsity and scaling. However, from the numerical experiments for random LSE problems, if the data is not either sparse or badly scaled, it is more suitable to adopt the normwise condition number to measure the conditioning of LSE since the explicit formula of the normwise condition number is more compact.