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.
Categories: math.FA math.OA math.SP
