generalized singular values
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2016 ◽  
Vol 33 (04) ◽  
pp. 1650023
Author(s):  
Yang Liu

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.


2010 ◽  
Vol 208 (1) ◽  
pp. 5-29 ◽  
Author(s):  
Götz Alefeld ◽  
Rolf Hoffmann ◽  
Günter Mayer ◽  
G. Wildenhain

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