Bounds on Perturbations of Generalized Singular Values and of Associated Subspaces

1993 ◽  
Vol 14 (1) ◽  
pp. 195-234 ◽  
Author(s):  
Ren-Cang Li
Keyword(s):  
Singular Values ◽  
Generalized Singular Values

On Low Rank Approximation of Linear Operators in p-Norms and Some Algorithms

2016 ◽  
Vol 33 (04) ◽  
pp. 1650023
Author(s):  
Yang Liu
Keyword(s):  
Linear Operator ◽  
Sparse Matrix ◽  
Matrix Completion ◽  
Singular Values ◽  
Linear Operators ◽  
Low Rank ◽  
Minkowski Distance ◽  
Low Rank Approximation ◽  
Rank Approximation ◽  
Generalized Singular Values

In this paper, we study the optimal or best approximation of any linear operator by low rank linear operators, especially, any linear operator on the [Formula: see text]-space, [Formula: see text], under [Formula: see text] norm, or in Minkowski distance. Considering generalized singular values and using techniques from differential geometry, we extend the classical Schmidt–Mirsky theorem in the direction of the [Formula: see text]-norm of linear operators for some [Formula: see text] values. Also, we develop and provide algorithms for finding the solution to the low rank approximation problems in some nontrivial scenarios. The results can be applied to, in particular, matrix completion and sparse matrix recovery.

Verification Algorithms for Generalized Singular Values

2010 ◽  
Vol 208 (1) ◽  
pp. 5-29 ◽  
Author(s):  
Götz Alefeld ◽  
Rolf Hoffmann ◽  
Günter Mayer ◽  
G. Wildenhain
Keyword(s):  
Singular Values ◽  
Verification Algorithms ◽  
Generalized Singular Values
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