Restricted Invertibility Revisited

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.

Dedicated to Jirka Matoušek