Forcing sparsity by projecting with respect to a non-diagonally weighted frobenius norm

1983 ◽  
Vol 25 (1) ◽  
pp. 125-129 ◽  
Author(s):  
R. B. Schnabel ◽  
Ph. L. Toint
Keyword(s):  
Frobenius Norm ◽  
Weighted Frobenius Norm

Partial Bi-Invariance of SE(3) Metrics1

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Gregory S. Chirikjian
Keyword(s):  
Rigid Body ◽  
Reference Frames ◽  
Distance Functions ◽  
Frobenius Norm ◽  
Transformation Matrices ◽  
Invariance Properties ◽  
Full Picture ◽  
Weighted Frobenius Norm ◽  
Rigid Body Motions ◽  
The Difference

In a flurry of articles in the mid to late 1990s, various metrics for the group of rigid-body motions, SE(3), were introduced for measuring distance between any two reference frames or rigid-body motions. During this time, it was shown that one can choose a smooth distance function that is invariant under either all left shifts or all right shifts, but not both. For example, if one defines the distance between two reference frames to be an appropriately weighted Frobenius norm of the difference of the corresponding homogeneous transformation matrices, this will be invariant under left shifts by arbitrary rigid-body motions. However, this is not the full picture—other invariance properties exist. Though the Frobenius norm is not invariant under right shifts by arbitrary rigid-body motions, for an appropriate weighting it is invariant under right shifts by pure rotations. This is also true for metrics based on the Lie-theoretic logarithm. This paper goes further to investigate the full invariance properties of distance functions on SE(3), clarifying the full subsets of motions under which both left and right invariance is possible.

Partial Bi-Invariance of SE(3) Metrics

2014 ◽  
Author(s):  
Gregory S. Chirikjian
Keyword(s):  
Rigid Body ◽  
Reference Frames ◽  
Distance Functions ◽  
Frobenius Norm ◽  
Transformation Matrices ◽  
Invariance Properties ◽  
Full Picture ◽  
Weighted Frobenius Norm ◽  
Rigid Body Motions ◽  
The Difference

In a flurry of articles in the mid to late 1990s, various metrics for the group of rigid-body motions, SE(3), were introduced for measuring distance between any two reference frames or rigid-body motions. During this time it was shown that one can choose a smooth distance function that is invariant under either all left shifts or all right shifts, but not both. For example, if one defines the distance between two reference frames to be an appropriately weighted Frobenius norm of the difference of the corresponding homogeneous transformation matrices, this will be invariant under left shifts by arbitrary rigid-body motions. However, this is not the full picture — other invariance properties exist. Though the Frobenius norm is not invariant under right shifts by arbitrary rigid-body motions, for an appropriate weighting it is invariant under right shifts by pure rotations. This is also true for metrics based on the Lie-theoretic logarithm. This paper goes further to investigate the full invariance properties of distance functions on SE(3), clarifying the full subsets of motions under which both left and right invariance is possible.

Minimum Weighted Frobenius Norm Discrete-Time FIR Filter With Embedded Unbiasedness

2018 ◽  
Vol 65 (9) ◽  
pp. 1284-1288 ◽  
Author(s):  
Sung Hyun You ◽  
Choon Ki Ahn ◽  
Yuriy S. Shmaliy ◽  
Shunyi Zhao
Keyword(s):  
Discrete Time ◽  
Fir Filter ◽  
Frobenius Norm ◽  
Weighted Frobenius Norm

Diagonal quasi-Newton methods via least change updating principle with weighted Frobenius norm

2020 ◽  
Author(s):  
Wah June Leong ◽  
Sharareh Enshaei ◽  
Sie Long Kek
Keyword(s):  
Frobenius Norm ◽  
Newton Methods ◽  
Weighted Frobenius Norm ◽  
Quasi Newton

scholarly journals Normwise condition numbers of the indefinite least squares problem with multiple right-hand sides

2021 ◽  
Vol 7 (3) ◽  
pp. 3692-3700
Author(s):  
Limin Li ◽  
Keyword(s):  
Least Squares ◽  
Numerical Experiments ◽  
Upper Bounds ◽  
Condition Numbers ◽  
Frobenius Norm ◽  
Least Squares Problem ◽  
Right Hand ◽  
Weighted Frobenius Norm

<abstract><p>In this paper, we investigate the normwise condition numbers of the indefinite least squares problem with multiple right-hand sides with respect to the weighted Frobenius norm and $ 2 $-norm. The closed formulas or upper bounds for these condition numbers are presented, which extend the earlier work for the indefinite least squares problem with single right-hand side. Numerical experiments are performed to illustrate the tightness of the upper bounds.</p></abstract>

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.

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