Restricted Invertibility Revisited

@article{Naor2017RestrictedIR,
  title={Restricted Invertibility Revisited},
  author={Assaf Naor and Pierre Youssef},
  journal={arXiv: Functional Analysis},
  year={2017},
  pages={657-691}
}
Suppose that \(m,n \in \mathbb{N}\) and that \(A: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}\) is a linear operator. It is shown here that if \(k,r \in \mathbb{N}\) satisfy \(k <r\leqslant \mathbf{rank}(A)\) then there exists a subset σ ⊆ {1, …, m} with | σ | = k such that the restriction of A to \(\mathbb{R}^{\sigma } \subseteq \mathbb{R}^{m}\) is invertible, and moreover the operator norm of the inverse \(A^{-1}: A(\mathbb{R}^{\sigma }) \rightarrow \mathbb{R}^{m}\) is at most a constant… 
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