Invertibility of random submatrices via tail decoupling and a Matrix Chernoff Inequality

Abstract

Let $X$ be a $n\times p$ matrix with coherence $\mu (X)={\max }_{j\ne j^{\prime }}|X_j^t{X}_{j^{\prime }}|$. We present a simplified and improved study of the quasi-isometry property for most submatrices of $X$ obtained by uniform column sampling. Our results depend on $\mu (X)$, $\Vert X\Vert$ and the dimensions with explicit constants, which improve the previously known values by a large factor. The analysis relies on a tail decoupling argument, of independent interest, and a recent version of the Non-Commutative Chernoff inequality (NCCI).