Restricted invertibility revisited

Abstract: Suppose that $m,n\in \mathbb{N}$ and that $A:\mathbb{R}^m\to \mathbb{R}^n$ is a linear operator. It is shown here that if $k,r\in \mathbb{N}$ satisfy $k<r\le \mathrm{\bf rank(A)}$ then there exists a subset $\sigma\subseteq {1,\ldots,m}$ with $|\sigma|=k$ such that the restriction of $A$ to $\mathbb{R}^{\sigma}\subseteq \mathbb{R}^m$ is invertible, and moreover the operator norm of the inverse $A^{-1}:A(\mathbb{R}^{\sigma})\to \mathbb{R}^m$ 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.