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

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  • Yang Liu

    (Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA)

Abstract

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 ℓp-space, p ∈ [1,∞), under ℓp 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 ℓp-norm of linear operators for some p 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.

Suggested Citation

  • Yang Liu, 2016. "On Low Rank Approximation of Linear Operators in p-Norms and Some Algorithms," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(04), pages 1-12, August.
    • Handle: RePEc:wsi:apjorx:v:33:y:2016:i:04:n:s0217595916500238
    DOI: 10.1142/S0217595916500238
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