Randomized strong rank-revealing QR for column subset selection and low-rank matrix approximation

We discuss a randomized strong rank-revealing QR factorization that effectively reveals the spectrum of a matrix $\textbf{M}$. This factorization can be used to address problems such as selecting a subset of the columns of $\textbf{M}$, computing its low-rank approximation, estimating its rank, or approximating its null space. Given a random sketching matrix $\pmb{\Omega}$ that satisfies the $\epsilon$-embedding property for a subspace within the range of $\textbf{M}$, the factorization relies on selecting columns that allow to reveal the spectrum via a deterministic strong rank-revealing QR factorization of $\textbf{M}^{sk} = \pmb{\Omega}\textbf{M}$, the sketch of $\textbf{M}$. We show that this selection leads to a factorization with strong rank-revealing properties, making it suitable for approximating the singular values of $\textbf{M}$.