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- W2222890437 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 $$k <rleqslant mathbf{rank}(A)$$ 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." @default.
- W2222890437 created "2016-06-24" @default.
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- W2222890437 date "2017-01-01" @default.
- W2222890437 modified "2023-10-18" @default.
- W2222890437 title "Restricted Invertibility Revisited" @default.
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- W2222890437 doi "https://doi.org/10.1007/978-3-319-44479-6_27" @default.
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