Rank constrained matrix best approximation problem

Wang, Hongxing

doi:10.1016/j.aml.2015.06.009

Cited by 12 publications

(2 citation statements)

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“…Let us denote by R m×n r the set of all m × n matrices of rank at most r. Then on the basis of [83,30,52,12,87], the minimal Frobenius norm solution to the problem min…”

Section: σ (A)) and Writementioning

confidence: 99%

Optimal modeling of nonlinear systems Method of variable injections

Torokhti,

Soto-Quiros

2024

Proyecciones (Antofagasta)

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Our work addresses a development and justification of the new approach to the modeling of nonlinear systems. Let $\f$ be an unknown input-output map of the system with a random input and output $\y$ and $\x$, respectively. It is assumed that $\y$ and $\x$ are available and covariance matrices formed from $\y$ and $\x$ are known. We determine a model of $\f$ so that an associated error is minimized. To this end, the model $\ttt_p$ is constructed as a sum of $p+1$ particular parts, in the form $\ttt_p (\y) = \sum_{j=0}^{p}G_j H_j Q_j(\vv_j)$ where $G_j$ and $ H_j$, for $j=0,\ldots, p$, are matrices to be determined, and $\vv_j$, for $j=1,\ldots,p$, is a special random vector called the injection. We denote $\vv_0=\y$. Injections $\vv_1,\ldots, \vv_p$ are aimed to diminish the error associated with the proposed model $\ttt_p$. Further, $Q_j$ is a special transform aimed to facilitate the numerical realization of model $\ttt_p$. It is determined in the way allowing us to optimally determine $G_j$ and $ H_j$ as a solution of $p+1$ separate error minimization problems which are simpler than the original minimization problem. The empirical determination of injections $\vv_1,\ldots, \vv_p$ is considered. The proposed method has several degrees of freedom to diminish the associated error. They are `degree' $p$ of $\ttt_p$, choice of matrices $G_0, H_0, \ldots,$ $G_p, H_p$, dimensions of matrices $G_0, H_0, \ldots,$ $G_p, H_p$ and injections $\vv_1,\ldots, \vv_p$, respectively. Four numerical examples are provided. At the end, the open problem is formulated.

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“…Let us denote by R m×n r the set of all m × n matrices of rank at most r. Then on the basis of [83,30,52,12,87], the minimal Frobenius norm solution to the problem min…”

Section: σ (A)) and Writementioning

confidence: 99%

Optimal modeling of nonlinear systems Method of variable injections

Torokhti,

Soto-Quiros

2024

Proyecciones (Antofagasta)

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“…In the RHS of (18), only the last term depends on S j . Therefore, on the basis of [24,[58][59][60][61][62], the minimal Frobenius norm solution to the problem min…”

Section: Unitary Matrices Andmentioning

confidence: 99%

Best approximations of non-linear mappings: Method of optimal injections

Torokhti¹,

Soto-Quiros²

2018

Preprint

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While the theory of operator approximation with any given accuracy is well elaborated, the theory of best constrained constructive operator approximation is still not so well developed. Despite increasing demands from applications this subject is hardly tractable because of intrinsic difficulties in associated approximation techniques. This paper concerns the best constrained approximation of a non-linear operator in probability spaces. We propose and justify a new approach and technique based on the following observation. Methods for best approximation are aimed at obtaining the best solution within a certain class; the accuracy of the solution is limited by the extent to which the class is suitable. By contrast, iterative methods are normally convergent but the convergence can be quite slow. Moreover, in practice only a finite number of iteration loops can be carried out and therefore, the final approximate solution is often unsatisfactorily inaccurate. A natural idea is to combine the methods for best approximation and iterative techniques to exploit their advantageous features. Here, we present an approach which realizes this.The proposed approximating operator has several degrees of freedom to minimize the associated error. In particular, one of the specific features of the approximating technique we develop is special random vectors called injections. They are determined in the way that allows us to further minimize the associated error.

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