Restricted invertibility revisited

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 $\mathbf{r}\mathbf{a}\mathbf{n}\mathbf{k}(\mathbf{A})$$thenthereexistsasubset$\sigma\subseteq {1,\ldots,m}$with$|\sigma|=k$suchthattherestrictionof$A$to$$\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)$arethesingularvaluesof$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.