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Let $X$ be a $n\times p$ matrix with coherence $\mu(X)=\max_{j\neq j'}|X_j^tX_{j'}|$. We present a simplified and improved study of thequasi-isometry property for most submatrices of $X$ obtained by uniform columnsampling. Our results depend on $\mu(X)$, $\|X\|$ and the dimensions withexplicit constants, which improve the previously known values by a largefactor. The analysis relies on a tail decoupling argument, of independentinterest, and a recent version of the Non-Commutative Chernoff inequality(NCCI).

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