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

TL;DR

This paper provides an improved analysis of the quasi-isometry property for random submatrices of a given matrix, using tail decoupling and a matrix Chernoff inequality, with explicit constants depending on coherence and dimensions.

Contribution

It introduces a simplified, sharper study of submatrix invertibility, leveraging tail decoupling and NCCI, with explicit bounds that outperform previous results.

Findings

Improved bounds on submatrix invertibility depending on coherence and dimensions

Explicit constants provided for quasi-isometry properties

Novel tail decoupling approach of independent interest

Abstract

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 the quasi-isometry property for most submatrices of $X$ obtained by uniform column sampling. Our results depend on $\mu(X)$, $\|X\|$ 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).