Applications of statistical condition estimation to the solution of linear systems

Laub, Alan J.; Xia, Jianlin

doi:10.1002/nla.570

Cited by 19 publications

(8 citation statements)

References 32 publications

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“…For a detailed discussion of condition number estimation, see the survey paper by Higham [37] and the references therein. More recent work includes Laub and Xia [50].…”

Section: Condition Number Estimationmentioning

confidence: 99%

Subspace Iteration Randomization and Singular Value Problems

Gu¹

2015

SIAM J. Sci. Comput.

174

231

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Abstract. A classical problem in matrix computations is the efficient and reliable approximation of a given matrix by a matrix of lower rank. The truncated singular value decomposition (SVD) is known to provide the best such approximation for any given fixed rank. However, the SVD is also known to be very costly to compute. Among the different approaches in the literature for computing low-rank approximations, randomized algorithms have attracted researchers' recent attention due to their surprising reliability and computational efficiency in different application areas. Typically, such algorithms are shown to compute with very high probability low-rank approximations that are within a constant factor from optimal, and are known to perform even better in many practical situations. In this paper, we present a novel error analysis that considers randomized algorithms within the subspace iteration framework and show with very high probability that highly accurate low-rank approximations as well as singular values can indeed be computed quickly for matrices with rapidly decaying singular values. Such matrices appear frequently in diverse application areas such as data analysis, fast structured matrix computations and fast direct methods for large sparse linear systems of equations and are the driving motivation for randomized methods. Furthermore, we show that the low-rank approximations computed by these randomized algorithms are actually rank-revealing approximations, and the special case of a rank-1 approximation can also be used to correctly estimate matrix 2-norms with very high probability. Our numerical experiments are in full support of our conclusions.key words: low-rank approximation, randomized algorithms, singular values, standard Gaussian matrix.1. Introduction. Randomized algorithms have established themselves as some of the most competitive methods for rapid low-rank matrix approximation, which is vital in many areas of scientific computing, including principal component analysis [47,65] and face recognition [60,78], large scale data compression [21,22,35,56] and fast approximate algorithms for PDEs and integral equations [16,33,57,71,72,83,82]. In this paper, we consider randomized algorithms for low-rank approximations and singular value approximations within the subspace iteration framework, leading to results that simultaneously retain the reliability of randomized algorithms and the typical faster convergence of subspace iteration methods.

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“…For a detailed discussion of condition number estimation, see the survey paper by Higham [37] and the references therein. More recent work includes Laub and Xia [50].…”

Section: Condition Number Estimationmentioning

confidence: 99%

Subspace Iteration Randomization and Singular Value Problems

Gu¹

2015

SIAM J. Sci. Comput.

174

231

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“…For the partial mixed and componentwise condition numbers, we consider the SSCE method [19,27], which has been used to estimate the condition numbers for the linear systems, the LLS problem, the matrix equations et al (e.g., [4,12,28,29,30]). As done in the aforementioned references, we devise Algorithm 3 to estimate the condition numbers κ mILS (A, b) and κ cILS in (3.18) and (3.19).…”

Section: Estimating Condition Numbers Under ∞-Normmentioning

confidence: 99%

On the partial condition numbers for the indefinite least squares problem

Wang

2018

Applied Numerical Mathematics

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The condition number of a linear function of the indefinite least squares solution is called the partial condition number for the indefinite least squares problem. In this paper, based on a new and very general condition number which can be called the unified condition number, we first present an expression of the partial unified condition number when the data space is measured by a general weighted product norm. Then, by setting the specific norms and weight parameters, we obtain the expressions of the partial normwise, mixed and componentwise condition numbers. Moreover, the corresponding structured partial condition numbers are also taken into consideration when the problem is structured. Considering the connections between the indefinite and total least squares problems, we derive the (structured) partial condition numbers for the latter, which generalize the ones in the literature. To estimate these condition numbers effectively and reliably, the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are applied and three related algorithms are devised. Finally, the obtained results are illustrated by numerical experiments.

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“…The SCE, proposed by Kenny and Laub [23], is an efficient method for estimating the condition numbers for linear systems [25,26], linear least squares problems [24], the Tikhonov regularization problem [10], the total least squares problem [11], eigenvalue problems [28], roots of polynomials [27], etc. Diao et al [9,12,13] applied the SCE to the (generalized) Sylvester equations.…”

Section: Introductionmentioning

confidence: 99%

Structured condition numbers and small sample condition estimation of symmetric algebraic Riccati equations

Diao

Qiao

2017

Applied Mathematics and Computation

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This paper is devoted to a structured perturbation analysis of the symmetric algebraic Riccati equations by exploiting the symmetry structure. Based on the analysis, the upper bounds for the structured normwise, mixed and componentwise condition numbers are derived. Due to the exploitation of the symmetry structure, our results are improvements of the previous work on the perturbation analysis and condition numbers of the symmetric algebraic Riccati equations. Our preliminary numerical experiments demonstrate that our condition numbers provide accurate estimates for the change in the solution caused by the perturbations on the data. Moreover, by applying the small sample condition estimation method, we propose a statistical algorithm for practically estimating the condition numbers of the symmetric algebraic Riccati equations.

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Applications of statistical condition estimation to the solution of linear systems

Cited by 19 publications

References 32 publications

Subspace Iteration Randomization and Singular Value Problems

Subspace Iteration Randomization and Singular Value Problems

On the partial condition numbers for the indefinite least squares problem

Structured condition numbers and small sample condition estimation of symmetric algebraic Riccati equations

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