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

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 [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.