Limited memory BFGS algorithm for the matrix approximation problem in Frobenius norm

2020 ◽  
Vol 39 (2) ◽  
Author(s):  
Chungen Shen ◽  
Changxing Fan ◽  
Yunlong Wang ◽  
Wenjuan Xue
Keyword(s):  
Approximation Problem ◽  
Frobenius Norm ◽  
Matrix Approximation ◽  
Limited Memory ◽  
The Matrix

scholarly journals The core inverse and constrained matrix approximation problem

2020 ◽  
Vol 18 (1) ◽  
pp. 653-661 ◽  
Author(s):  
Hongxing Wang ◽  
Xiaoyan Zhang
Keyword(s):  
Unique Solution ◽  
Approximation Problem ◽  
Frobenius Norm ◽  
Matrix Approximation ◽  
The Core ◽  
Core Inverse

Abstract In this article, we study the constrained matrix approximation problem in the Frobenius norm by using the core inverse: ||Mx-b|{|}_{F}=\hspace{.25em}\min \hspace{1em}\text{subject}\hspace{.25em}\text{to}\hspace{1em}x\in {\mathcal R} (M), where M\in {{\mathbb{C}}}_{n}^{\text{CM}} . We get the unique solution to the problem, provide two Cramer’s rules for the unique solution and establish two new expressions for the core inverse.

scholarly journals A New Iterative Algorithm for Solving a Class of Matrix Nearness Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Xuefeng Duan ◽  
Chunmei Li
Keyword(s):  
Iterative Algorithm ◽  
Projection Algorithm ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Von Neumann ◽  
Numerical Examples ◽  
Matrix Nearness Problem ◽  
Least Frobenius Norm Solution ◽  
Alternating Projection ◽  
The Matrix

Based on the alternating projection algorithm, which was proposed by Von Neumann to treat the problem of finding the projection of a given point onto the intersection of two closed subspaces, we propose a new iterative algorithm to solve the matrix nearness problem associated with the matrix equations AXB=E, CXD=F, which arises frequently in experimental design. If we choose the initial iterative matrix X0=0, the least Frobenius norm solution of these matrix equations is obtained. Numerical examples show that the new algorithm is feasible and effective.

scholarly journals An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Feng Yin ◽  
Guang-Xin Huang
Keyword(s):  
Iterative Algorithm ◽  
Special Kind ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Sylvester Matrix ◽  
Special Cases ◽  
The Matrix ◽  
Solution Pair ◽  
Sylvester Matrix Equations ◽  
Matrix Pair

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations(AXB-CYD,EXF-GYH)=(M,N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matricesXandY. When the matrix equations are consistent, for any initial generalized reflexive matrix pair[X1,Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair[X̂,Ŷ]to a given matrix pair[X0,Y0]in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair[X̃*,Ỹ*]of a new corresponding generalized coupled Sylvester matrix equation pair(AX̃B-CỸD,EX̃F-GỸH)=(M̃,Ñ), whereM̃=M-AX0B+CY0D,Ñ=N-EX0F+GY0H. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

scholarly journals Low-rank matrix completion and denoising under Poisson noise

2020 ◽  
Author(s):  
Andrew D McRae ◽  
Mark A Davenport
Keyword(s):  
Matrix Completion ◽  
Universal Constant ◽  
Poisson Noise ◽  
Nuclear Norm ◽  
Low Rank ◽  
Frobenius Norm ◽  
Rank Matrix ◽  
The Matrix ◽  
Low Rank Matrix Completion ◽  
Low Rank Matrix

Abstract This paper considers the problem of estimating a low-rank matrix from the observation of all or a subset of its entries in the presence of Poisson noise. When we observe all entries, this is a problem of matrix denoising; when we observe only a subset of the entries, this is a problem of matrix completion. In both cases, we exploit an assumption that the underlying matrix is low-rank. Specifically, we analyse several estimators, including a constrained nuclear-norm minimization program, nuclear-norm regularized least squares and a non-convex constrained low-rank optimization problem. We show that for all three estimators, with high probability, we have an upper error bound (in the Frobenius norm error metric) that depends on the matrix rank, the fraction of the elements observed and the maximal row and column sums of the true matrix. We furthermore show that the above results are minimax optimal (within a universal constant) in classes of matrices with low-rank and bounded row and column sums. We also extend these results to handle the case of matrix multinomial denoising and completion.

Eigenvalue localization for complex matrices

2014 ◽  
Vol 27 ◽  
Author(s):  
Ibrahim Gumus ◽  
Omar Hirzallah ◽  
Fuad Kittaneh
Keyword(s):  
Frobenius Norm ◽  
Complex Matrix ◽  
Complex Matrices ◽  
Eigenvalue Localization ◽  
The Matrix

Let $A$ be an $n\times n$ complex matrix with $n\geq 3$. It is shown that at least $n-2$ of the eigenvalues of $A$ lie in the disk \begin{equation*}\left\vert z-\frac{\func{tr}A}{n}\right\vert \leq \sqrt{\frac{n-1}{n}\left(\sqrt{\left( \left\Vert A\right\Vert _{2}^{2}-\frac{\left\vert \func{tr} A\right\vert ^{2}}{n}\right) ^{2}-\frac{\left\Vert A^{\ast }A-AA^{\ast}\right\Vert _{2}^{2}}{2}}-\frac{\limfunc{spd}\nolimits^{2}(A)}{2}\right) },\end{equation*} where $\left\Vert A\right\Vert _{2},$ $\func{tr}A$, and $\limfunc{spd}(A)$ denote the Frobenius norm, the trace, and the spread of $A$, respectively. In particular, if $A=\left[ a_{ij}\right] $ is normal, then at least $n-2$ of the eigenvalues of $A$ lie in the disk {\small \begin{eqnarray*} & & \left\vert z-\frac{\func{tr}A}{n}\right\vert \\ & & \leq \sqrt{\frac{n-1}{n}\left( \frac{\left\Vert A\right\Vert _{2}^{2}}{2}-\frac{\left\vert \func{tr}A\right\vert ^{2}}{n}-\frac{3}{2}\max_{i,j=1,\dots,n} \left( \sum_{\substack{ k=1 \\ k\neq i}}^{n}\left\vert a_{ki}\right\vert ^{2}+\sum_{\substack{ k=1 \\ k\neq j}}^{n}\left\vert a_{kj}\right\vert ^{2}+\frac{\left\vert a_{ii}-a_{jj}\right\vert ^{2}}{2}\right) \right) }. \end{eqnarray*}} Moreover, the constant $\frac{3}{2}$ can be replaced by $4$ if the matrix $A$ is Hermitian.

Sign in / Sign up

Export Citation Format

Share Document