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Restricted Invertibility Revisited

  • Assaf Naor
  • Pierre Youssef
Chapter

Abstract

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 multiple of the quantity \(\sqrt{mr/((r - k)\sum _{i=r }^{m }\mathsf{s } _{i } (A)^{2 } )}\), where \(\mathsf{s}_{1}(A)\geqslant \ldots \geqslant \mathsf{s}_{m}(A)\) are the singular values of A. This improves over a series of works, starting from the seminal Bourgain–Tzafriri Restricted Invertibility Principle, through the works of Vershynin, Spielman–Srivastava and Marcus–Spielman–Srivastava. In particular, this directly implies an improved restricted invertibility principle in terms of Schatten–von Neumann norms.

Notes

Acknowledgements

We thank Bill Johnson for helpful discussions. This work was initiated while we were participating in the workshop Beyond Kadison–Singer: paving and consequences at the American Institute of Mathematics. We thank the organizers for the excellent working conditions.

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© Springer International publishing AG 2017

Authors and Affiliations

  1. 1.Mathematics DepartmentPrinceton University Fine HallPrincetonUSA
  2. 2.Laboratoire de Probabilités et de Modèles AléatoiresUniversité Paris-DiderotParisFrance

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