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

TL;DR

This work addresses the problem of revealing a matrix spectrum while enabling efficient column subset selection and low-rank approximation. It introduces a randomized strong RRQR algorithm that uses a sketch $\mathbf{M}^{sk}=\Omega\mathbf{M}$ to select pivot columns via a deterministic SRQR on the sketch, followed by a QR without pivoting on the permuted original matrix. The authors prove that the randomized construction preserves the strong rank-revealing properties up to a factor $\tilde f$, yielding reliable approximations to leading and trailing singular values with reduced computational and communication costs. Practically, this enables fast low-rank approximations, rank estimation, and null-space computations on large matrices, with competitive accuracy compared to classical QRCP and randomized SVD methods. Numerical experiments demonstrate substantial speedups (e.g., up to $15.9\times$) while maintaining accurate spectral information across diverse test matrices.

Abstract

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 $\pmbΩ$ that satisfies the $ε$-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 $\textbf{M}^{sk} = \pmbΩ\textbf{M}$, 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 $\textbf{M}$.

Figures (11)

• Figure 1: parallelogram OABC
• Figure 2: volume of a $3\times3$ matrix $( {\textbf{v}}_1 \ ,\ {\textbf{v}}_2 \ ,\ {\textbf{v}}_3)$
• Figure 3: Volume of the sketch of an orthogonal matrix matrix obtained from sampling columns of the identity matrix.
• Figure 4: Ratios $\sigma_i({\textbf{M}})/\sigma_i({\textbf{R}}_{11})$ of $8192\times500$ Kahan matrix
• Figure 5: Singular values and L/R-values for $8192\times500$ Kahan matrix

Theorems & Definitions (28)

• Theorem 2.1: Existence of a strong RRQR factorization srrqr
• Remark 2.1
• Definition 2.1: $\epsilon$-embedding
• Definition 2.2
• Theorem 2.2
• proof
• Lemma 2.1: Sketched least squares problem
• proof
• Definition 3.1: see, e.g. Page 154 of strang2006linear
• Lemma 3.1